❩❇■❘❑❆ ◆❆▲❖● ■❩

❱❊❑❚❖❘❙❑■❍ P❘❖❙❚❖❘❖❱

❆❥❞❛ ✐♥ ▼❛❥❛

❋❖➆◆❊❘

■③❞❛❥❛t❡❧❥ ✐♥ ③❛❧♦➸♥✐❦✿

❋❛❦✉❧t❡t❛ ③❛ ❧♦❣✐st✐❦♦ ❯♥✐✈❡r③❡ ✈ ▼❛r✐❜♦r✉

❆✈t♦r✿

❆❥❞❛ ❋♦➨♥❡r ✐♥ ▼❛❥❛ ❋♦➨♥❡r

▲❡❦t♦r✐r❛♥❥❡✿

❆♥❞r❡❥❛ ❷✉r✐♥

❈■P ✲ ❑❛t❛❧♦➸♥✐ ③❛♣✐s ♦ ♣✉❜❧✐❦❛❝✐❥✐

◆❛r♦❞♥❛ ✐♥ ✉♥✐✈❡r③✐t❡t♥❛ ❦♥❥✐➸♥✐❝❛✱ ▲❥✉❜❧❥❛♥❛

✺✶✷✳✻✹✷✭✵✼✺✳✽✮✭✵✼✻✳✶✮

❋❖➆◆❊❘✱ ❆❥❞❛

❩❜✐r❦❛ ♥❛❧♦❣ ✐③ ✈❡❦t♦rs❦✐❤ ♣r♦st♦r♦✈ ❬❊❧❡❦tr♦♥s❦✐ ✈✐r❪ ✴

❆❥❞❛ ✐♥ ▼❛❥❛ ❋♦➨♥❡r✳ ✲ ❈❡❧❥❡ ✿ ❋❛❦✉❧t❡t❛ ③❛ ❧♦❣✐st✐❦♦✱ ✷✵✵✼

◆❛↔✐♥ ❞♦st♦♣❛ ✭❯❘▲✮✿ ❤tt♣✿✴✴✢✳✉♥✐✲♠❜✳s✐✴❡❦♥❥✐❣❡✴❩❜✐r❦❛❴♥❛❧♦❣❴✐③❴✈❡❦t♦rs❦✐❤❴♣r♦st♦r♦✈✳♣❞❢✳

✲ ❖♣✐s t❡♠❡❧❥✐ ♥❛ ✈❡r③✐❥✐ ③ ❞♥❡ ✶✹✳✵✽✳✷✵✵✼

■❙❇◆ ✾✼✽✲✾✻✶✲✻✺✻✷✲✵✾✲✻

✶✳ ❋♦➨♥❡r✱ ▼❛❥❛

✷✸✹✺✹✵✽✵✵

❑❛③❛❧♦

✶ ❱❡❦t♦rs❦✐ ♣r♦st♦r✐

✹

✷ ❇❛♥❛❝❤♦✈✐ ♣r♦st♦r✐

✶✹

✸ ❍✐❧❜❡rt♦✈✐ ♣r♦st♦r✐

✸✶

✸

P♦❣❧❛✈❥❡ ✶

❱❡❦t♦rs❦✐ ♣r♦st♦r✐

✶✳ ◆❛❥ ❜♦ V ♣r♦st♦r r❡❛❧♥✐❤ ③❛♣♦r❡❞✐❥ ✐♥ U ♠♥♦➸✐❝❛ t❛❦✐❤ r❡❛❧♥✐❤ ③❛♣♦r❡❞✐❥✱

❦✐ ✐♠❛❥♦ ❧✐❤❡ ↔❧❡♥❡ ❡♥❛❦❡ ♥✐↔✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ U ♣♦❞♣r♦st♦r ♣r♦st♦r❛ V

✐♥ ♣♦✐➨↔✐t❡ ❦❛❦ ♥❥❡❣♦✈ ❦♦♠♣❧❡♠❡♥t✳

◆❛♠✐❣✳ ❑♦♠♣❧❡♠❡♥t ♣r♦st♦r❛ U ❥❡ ♥❛ ♣r✐♠❡r

W = {{xn}n∈N | x2n = 0, n ∈ N}.

✷✳ P♦✐➨↔✐t❡ ❦❛❦♦ ♠♥♦➸✐❝♦ ❧✐♥❡❛r♥♦ ♥❡♦❞✈✐s♥✐❤ ❡❧❡♠❡♥t♦✈ ♣r♦st♦r❛ R3✱ ❦✐

③❛❞♦➨↔❛❥♦ r❡➨✐t✈✐ ❡♥❛↔❜❡ x1 − x2 + x3 = 0✳

❘❡➨✐t❡✈✳ ■s❦❛♥❛ ♠♥♦➸✐❝❛ ❥❡ ♥❛ ♣r✐♠❡r {(1, 1, 0), (0, 1, 1), (2, 1, −1)}✳

✸✳ ❩❛ ✈s❛❦ n ∈ N ♥❛❥ ❜♦

xn = (1, . . . , 1, 0, 0, . . .).

n

❆❧✐ ❥❡ ♠♥♦➸✐❝❛ ③❛♣♦r❡❞✐❥ {xn}n∈N ❧✐♥❡❛r♥♦ ♥❡♦❞✈✐s♥❛❄

❘❡➨✐t❡✈✳ ▼♥♦➸✐❝❛ ③❛♣♦r❡❞✐❥ {xn}n∈N ❥❡ ❧✐♥❡❛r♥♦ ♥❡♦❞✈✐s♥❛✳

✹

✹✳ ◆❛❥ ❜♦❞♦ ✈❡❦t♦r❥✐ b1, b2, . . . , bn+1 ❧✐♥❡❛r♥❛ ❦♦♠❜✐♥❛❝✐❥❛ ❧✐♥❡❛r♥♦ ♥❡♦❞✲

✈✐s♥✐❤ ✈❡❦t♦r❥❡✈ a1, a2, . . . , an✳ P♦❦❛➸✐t❡✱ ❞❛ s♦ ✈❡❦t♦r❥✐ b1, b2, . . . , bn+1

❧✐♥❡❛r♥♦ ♦❞✈✐s♥✐✳

◆❛♠✐❣✳ ❱❡❦t♦r❥❡ b1, b2, . . . , bn+1 ③❛♣✐➨✐t❡ ❦♦t ❧✐♥❡❛r♥♦ ❦♦♠❜✐♥❛❝✐❥♦ ✈❡❦✲

t♦r❥❡✈ a1, a2, . . . , an✳ ❙ ♣♦♠♦↔❥♦ t❡❣❛ ③❛♣✐s❛ ♣♦❦❛➸✐t❡ ➸❡❧❡♥♦✳

✺✳ P♦❦❛➸✐t❡✱ ❞❛ s♦ ❢✉♥❦❝✐❥❡ ex, e−x ✐♥ sin x ❧✐♥❡❛r♥♦ ♥❡♦❞✈✐s♥❡ ✈ ♣r♦st♦r✉ RR✳

◆❛♠✐❣✳ ❩ ♦♣❛③♦✈❛♥❥❡♠ ✈r❡❞♥♦st✐ ❢✉♥❦❝✐❥ ✈ x = 0 ✐♥ x = π ♣♦❦❛➸❡♠♦

➸❡❧❡♥♦✳

✻✳ ❆❧✐ s♦ ❢✉♥❦❝✐❥❡ 2✱ cos 2x ✐♥ cos2 x ❧✐♥❡❛r♥♦ ♦❞✈✐s♥❡❄

✼✳ ❉♦❧♦↔✐t❡ t ∈ R t❛❦♦✱ ❞❛ ❜♦ ♠♥♦➸✐❝❛

U = {(x, y, z) ∈ R3 | x − t(y + 2z − 2) = 4}

✈❡❦t♦rs❦✐ ♣♦❞♣r♦st♦r ♣r♦st♦r❛ R3✳ P♦✐➨↔✐t❡ ❦❛❦♦ ♠❛❦s✐♠❛❧♥♦ ❧✐♥❡❛r♥♦

♥❡♦❞✈✐s♥♦ ♣♦❞♠♥♦➸✐❝♦ t❡❣❛ ♣r♦st♦r❛✳

◆❛♠✐❣✳ ■③❦❛➸❡ s❡✱ ❞❛ ❥❡ t = 2✳ ■s❦❛♥❛ ♠♥♦➸✐❝❛ ❥❡ ♥❛ ♣r✐♠❡r {(4, 0, 1), (2, 1, 0)}✳

✽✳ ◆❛❥ ❜♦ V = R4[x] ♣r♦st♦r r❡❛❧♥✐❤ ♣♦❧✐♥♦♠♦✈ st♦♣♥❥❡ ♥❛❥✈❡↔ 4 ✐♥ ♥❛❥

❜♦st❛

U = {p ∈ V | p′′ = 0} ✐♥ W = L{1 + x, x + x2, x2 + x3, x3 + x4},

❦❥❡r ❥❡ p′′ ❞r✉❣✐ ♦❞✈♦❞ ♣♦❧✐♥♦♠❛ p✳ ❆❧✐ ❥❡ U ⊆ W ?

◆❛♠✐❣✳ ❖♣❛③✐♠♦✱ ❞❛ ❥❡ U = L{1, x}✳ ❑❡r 1, x /∈ W ✱ ❥❡ ♦❞❣♦✈♦r ♥❛

③❛st❛✈❧❥❡♥♦ ✈♣r❛➨❛♥❥❡ ♥❡❣❛t✐✈❡♥✳

✺

✾✳ ◆❛❥ ❜♦ V = R3[x] ♣r♦st♦r r❡❛❧♥✐❤ ♣♦❧✐♥♦♠♦✈ st♦♣♥❥❡ ♥❛❥✈❡↔ 3✳ ❉♦❧♦↔✐t❡

α, β ∈ R t❛❦♦✱ ❞❛ ❜♦st❛ ♠♥♦➸✐❝✐ U = {p ∈ V | p(0)−(α−1)p′(0)+α = β}

✐♥ W = {p ∈ V | p(0)−p′′(0) = β +1, p(−1) = 0} ✈❡❦t♦rs❦❛ ♣♦❞♣r♦st♦r❛

♣r♦st♦r❛ V ✳ P♦✐➨↔✐t❡ ❜❛③♦ ♣r❡s❡❦❛ U ∩ W ✐♥ ❜❛③♦ ✈s♦t❡ U + W ✳

◆❛♠✐❣✳ ■③❦❛➸❡ s❡✱ ❞❛ ❥❡ α = β = −1✳ ❇❛③❛ ♣r❡s❡❦❛ ❥❡ BU∩W = {2 − x +

x2 + 4x3}, ❜❛③❛ ✈s♦t❡ ❥❡ BU+W = {1, x, x2, x3}.

✶✵✳ ❑❛❦➨♥❛ ♠♦r❛ ❜✐t✐ ❢✉♥❦❝✐❥❛ f : R → R✱ ❞❛ ❜♦ ♠♥♦➸✐❝❛ ❢✉♥❦❝✐❥ {1, f, f2, f3, . . .}

❧✐♥❡❛r♥♦ ♦❞✈✐s♥❛❄

❘❡➨✐t❡✈✳ ❖❜st❛❥❛❥♦ t❛❦✐ α0, α1, . . . , αn ∈ R✱ ♥❡ ✈s✐ ♥✐↔✱ ❞❛ ❥❡

α0 + α1f(x) + α2f(x)2 + . . . + αnf(x)n = 0

③❛ ✈s❡ x ∈ R✳ ❩❛♣✐s❛♥♦ ❞r✉❣❛↔❡✱ ✈s❛❦❛ ❢✉♥❦❝✐❥s❦❛ ✈r❡❞♥♦st ❥❡ ♥✐↔❧❛

✭♥❡♥✐↔❡❧♥❡❣❛✮ ♣♦❧✐♥♦♠❛

p(y) = α0 + α1y + α2y2 + . . . + αnyn.

❚♦❞❛ ♣♦❧✐♥♦♠ ✐♠❛ ❧❡ ❦♦♥↔♥♦ ♠♥♦❣♦ ♥✐↔❡❧✳ ❚♦r❡❥ ❥❡ ♣♦tr❡❜❡♥ ♣♦❣♦❥ ③❛

❧✐♥❡❛r♥♦ ♦❞✈✐s♥♦st ♠♥♦➸✐❝❡ {1, f, f2, f3, . . .}✱ ❞❛ f ③❛✈③❛♠❡ ❧❡ ❦♦♥↔♥♦

♠♥♦❣♦ r❛③❧✐↔♥✐❤ ✈r❡❞♥♦st✐✳ ❚❛ ♣♦❣♦❥ ♣❛ ❥❡ t✉❞✐ ③❛❞♦st❡♥✳ ◆❛♠r❡↔✱ ↔❡ f

③❛✈③❛♠❡ ❧❡ ✈r❡❞♥♦st✐ c1, c2, . . . , cm ∈ R✱ ❥❡

(f (x) − c1) · . . . · (f(x) − cm) = 0

③❛ ✈s❛❦ x ∈ R✱ t♦r❡❥

f (x)m + αm−1f(x)m−1 + . . . + α1f(x) + α0 = 0,

❦❥❡r ❥❡ α0 = (−1)mc1 · . . . · cm ∈ R✱ . . .✱ αm−1 = −(c1 + . . . + cm) ∈ R✳

P♦s❧❡❞✐❝❛ s❡✈❡❞❛ ❥❡✱ ❞❛ ③✈❡③♥❛ ♥❡❦♦♥st❛♥t♥❛ ❢✉♥❦❝✐❥❛ f ♥✐♠❛ t❡ ❧❛st♥♦st✐✳

✶✶✳ ◆❛❥ ❜♦ A = Mn(R) ♣r♦st♦r ✈s❡❤ r❡❛❧♥✐❤ ♠❛tr✐❦ ✐♥ ♥❛❥ ❜♦st❛

S = {A ∈ A | AT = A} t❡r K = {A ∈ A | AT = −A},

♣r✐ ↔❡♠❡r AT ♦③♥❛↔✉❥❡ tr❛♥s♣♦♥✐r❛♥♦ ♠❛tr✐❦♦ ♠❛tr✐❦❡ A✳

✻

✭❛✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ A = S ⊕ K✳

✭❜✮ ❩❛♣✐➨✐t❡ ❜❛③✐ ♣r♦st♦r♦✈ S ✐♥ K✳

❘❡➨✐t❡✈✳

✭❛✮ ❱s❛❦ A ∈ A ❧❛❤❦♦ ③❛♣✐➨❡♠♦ ✈ ♦❜❧✐❦✐

A + AT

A − AT

A =

+

,

2

2

❦❥❡r ❥❡

A + AT ∈ S ✐♥ A − AT ∈ K.

2

2

❷❡ ❥❡ A ∈ S ∩ K✱ ❥❡ AT = A = −A✱ ✐③ ↔❡s❛r s❧❡❞✐ A = 0✳ ❚♦r❡❥ ❥❡

S ⊕ K = {0}✳

✭❜✮ ❇❛③❛ ✐♥ ❞✐♠❡♥③✐❥❛ ♣r♦st♦r❛ S✿

BS = {Eii | 1 ≤ i ≤ n} ∪ {Eij + Eji | i = j, 1 ≤ i, j ≤ n, },

❞✐♠

n(n + 1)

S =

.

2

❇❛③❛ ✐♥ ❞✐♠❡♥③✐❥❛ ♣r♦st♦r❛ K✿

BK = {Eij − Eji | i = j, 1 ≤ i, j ≤ n, },

❞✐♠

n(n − 1)

K =

.

2

✶✷✳ ◆❛❥ ❜♦ Tn(R) ♣r♦st♦r ③❣♦r❛❥ tr✐❦♦t♥✐❤ n × n ♠❛tr✐❦✱ Sn(R) ♣❛ ♣r♦st♦r

str♦❣♦ s♣♦❞❛❥ tr✐❦♦t♥✐❤ n×n ♠❛tr✐❦✳ P♦✐➨↔✐t❡ ❦❛❦♦ ❜❛③♦ ♣r♦st♦r♦✈ Tn(R)

✐♥ Sn(R) t❡r ③❛♣✐➨✐t❡ ♥❥✉♥✐ ❞✐♠❡♥③✐❥✐✳

❘❡➨✐t❡✈✳ ❇❛③❛ ✐♥ ❞✐♠❡♥③✐❥❛ ♣r♦st♦r❛ Tn(R)✿

BTn(R) = {Eii | 1 ≤ i ≤ n} ∪ {Eij | 1 ≤ i ≤ n, i < j ≤ n},

❞✐♠

(n + 1)(n + 2)

Tn(R) =

.

2

✼

❇❛③❛ ✐♥ ❞✐♠❡♥③✐❥❛ ♣r♦st♦r❛ Sn(R)✿

BSn(R) = {Eij | 1 < i ≤ n, 1 ≤ j < i},

❞✐♠

n(n − 1)

Sn(R) =

.

2

✶✸✳ ❉❡✜♥✐r❛❥♠♦ ♣r❡s❧✐❦❛✈❡ A, B, C : R[0,1] → R[0,1] s ♣r❡❞♣✐s✐

✭❛✮ (Af)(x) = 3 f(x)✱

✭❜✮ (Bf)(x) = f(1 − x)✱

✭❝✮ (Cf)(x) = xf(x)✳

❆❧✐ s♦ ♣r❡s❧✐❦❛✈❡ ❧✐♥❡❛r♥❡❄ ❉♦❧♦↔✐t❡ ❥❡❞r♦ ✐♥ ③❛❧♦❣♦ ✈r❡❞♥♦st✐ ❧✐♥❡❛r♥✐❤

♣r❡s❧✐❦❛✈✳

❘❡➨✐t❡✈✳ Pr❡s❧✐❦❛✈✐ B ✐♥ C st❛ ❧✐♥❡❛r♥✐✳ Pr❡s❧✐❦❛✈❛ B ❥❡ ❜✐❥❡❦t✐✈♥❛✳

◆✐ t❡➸❦♦ ♣r❡✈❡r✐t✐✱ ❞❛ ❥❡ KerC = {f ∈ R[0,1] | f(x) = 0, x ∈ (0, 1]} ✐♥

ImC = {f ∈ R[0,1] | f(0) = 0}✳

✶✹✳ ◆❛❥ ❜♦ A ❧✐♥❡❛r❡♥ ♦♣❡r❛t♦r ♥❛ ✈❡❦t♦rs❦❡♠ ♣r♦st♦r✉ V ✳ P♦❦❛➸✐t❡✿ A ❥❡

✐♥❥❡❦t✐✈❡♥ ♥❛t❛♥❦♦ t❡❞❛❥✱ ❦♦ s❧✐❦❛ ❧✐♥❡❛r♥♦ ♥❡♦❞✈✐s♥❡ ♠♥♦➸✐❝❡ ✈ ❧✐♥❡❛r♥♦

♥❡♦❞✈✐s♥❡ ♠♥♦➸✐❝❡✳

❘❡➨✐t❡✈✳ (=⇒) ◆❛❥ ❜♦ S = {s1, s2, . . . , sn} ❧✐♥❡❛r♥♦ ♥❡♦❞✈✐s♥❛ ♣♦❞♠✲

♥♦➸✐❝❛ ♣r♦st♦r❛ V ✳ P♦t❡♠ ❥❡ 0 = α1As1 + α2As2 + . . . + αnAsn =

A(α1s1 + α2s2 + . . . + αnsn)✳ ❑❡r ❥❡ ♣♦ ♣r❡❞♣♦st❛✈❦✐ A ✐♥❥❡❦t✐✈❡♥ ♦♣✲

❡r❛t♦r✱ s❧❡❞✐ α1s1 + α2s2 + . . . + αnsn = 0✱ ❦❛r ♥❛s ♣r✐✈❡❞❡ ❞♦ ➸❡❧❡♥❡❣❛

r❡③✉❧t❛t❛ α1 = α2 = . . . = αn = 0✳

(⇐=) Pr❡❞♣♦st❛✈✐♠♦✱ ❞❛ A ♥✐ ✐♥❥❡❦t✐✈❡♥ ♦♣❡r❛t♦r✳ P♦t❡♠ ♦❜st❛❥❛ t❛❦

0 = x ∈ V ✱ ❞❛ ❥❡ Ax = 0✳ ❑❡r ❥❡ {x} ❧✐♥❡❛r♥♦ ♥❡♦❞✈✐s♥❛ ♠♥♦➸✐❝❛✱ ❥❡

t✉❞✐ ♠♥♦➸✐❝❛ {Ax} = {0} ❧✐♥❡❛r♥♦ ♥❡♦❞✈✐s♥❛✱ ❦❛r ♣❛ ♥✐ r❡s✳

✶✺✳ ◆❛❥ ❜♦st❛ U ✐♥ V ❦♦♥↔♥♦ r❛③s❡➸♥❛ ✈❡❦t♦rs❦❛ ♣r♦st♦r❛ ♥❛❞ ♣♦❧❥❡♠ F ✐♥

♥❛❥ ❜♦ A : U → V ❧✐♥❡❛r❡♥ ♦♣❡r❛t♦r✳ P♦❦❛➸✐t❡✿ A ❥❡ s✉r❥❡❦t✐✈❡♥ ♥❛t❛♥❦♦

t❡❞❛❥✱ ❦♦ s❧✐❦❛ ♦❣r♦❞❥❡ ✈ ♦❣r♦❞❥❡✳

✽

❘❡➨✐t❡✈✳ (=⇒) ◆❛❥ ❜♦ S = {s1, s2, . . . , sn} ♦❣r♦❞❥❡ ♣r♦st♦r❛ U ✐♥ ♥❛❥

❜♦ v ∈ V ✳ P♦t❡♠ ♦❜st❛❥❛ t❛❦ u ∈ U✱ ❞❛ ❥❡ v = Au = A(α1s1 + α2s2 +

. . . + αnsn) = α1(As1) + α2(As2) + . . . + αn(Asn) ③❛ ♥❡❦❡ αi ∈ F.

(⇐=) ❷❡ ❥❡ ♠♥♦➸✐❝❛ S ♦❣r♦❞❥❡ ♣r♦st♦r❛ U✱ ❥❡ A(S) = {As | s ∈ S}

♦❣r♦❞❥❡ ♣r♦st♦r❛ V ✳ ◆❛❥ ❜♦ v ∈ V = L(A(S))✳ P♦t❡♠ ❥❡ v = α1As1 +

α2As2 + . . . + αnAsn = A(α1s1 + α2s2 + . . . + αnsn) = Au, si ∈ S✱ αj ∈ F ✱

1 ≤ i, j ≤ n✳

✶✻✳ ◆❛❥ ❜♦❞♦ x, y, z ∈ R✳ ❆❧✐ ❥❡ r❛✈♥✐♥❛ z = x + y ❤✐♣❡rr❛✈♥✐♥❛❄ P♦✐➨↔✐t❡

t❛❦❡ ❧✐♥❡❛r♥❡ ❢✉♥❦❝✐♦♥❛❧❡ f✱ ❞❛ ❥❡ Kerf t❛ r❛✈♥✐♥❛✳

❘❡➨✐t❡✈✳ ❘❛✈♥✐♥❛ z = x + y ❥❡ ❤✐♣❡rr❛✈♥✐♥❛✳ ◆❛❥ ❜♦ a ∈ R ✐♥ fa

♣r❡s❧✐❦❛✈❛ ❞❡✜♥✐r❛♥❛ s ♣r❡❞♣✐s♦♠ fa(x, y, z) = a(x + y − z)✳ ❏❡❞r♦ t❛❦✐❤

♣r❡s❧✐❦❛✈ ❥❡ r❛✈♥✐♥❛ z = x + y✳

✶✼✳ ◆❛❥ ❜♦ V ♣r♦st♦r ♣♦❧✐♥♦♠♦✈ ③ r❡❛❧♥✐♠✐ ❦♦❡✜❝✐❡♥t✐ st♦♣♥❥❡ ♠❛♥❥➨❡ ❛❧✐

❡♥❛❦❡ n ✐♥ ♥❛❥ ❜♦ H = {a1x+. . .+anxn | ai ∈ R}✳ ❆❧✐ ❥❡ H ❤✐♣❡rr❛✈♥✐♥❛❄

P♦✐➨↔✐t❡ ❦❛❦ ❧✐♥❡❛r♥✐ ❢✉♥❦❝✐♦♥❛❧ f ③ ❧❛st♥♦st❥♦ Kerf = H ✳

❘❡➨✐t❡✈✳ ▼♥♦➸✐❝❛ H ❥❡ ❤✐♣❡rr❛✈♥✐♥❛✳ ❷❡ ❞❡✜♥✐r❛♠♦ ♣r❡s❧✐❦❛✈♦ f : V →

R s ♣r❡❞♣✐s♦♠ f (p) = p(0)✱ p ∈ V ✱ ♣♦t❡♠ ❥❡ Kerf = H✳

✶✽✳ ◆❛❥ ❜♦ V ❦♦♥↔♥♦ r❛③s❡➸❡♥ ✈❡❦t♦rs❦✐ ♣r♦st♦r ♥❛❞ ♣♦❧❥❡♠ F ✐♥ ♥❛❥ ❜♦st❛ f

t❡r g ♥❡♥✐↔❡❧♥❛ ❧✐♥❡❛r♥❛ ❢✉♥❦❝✐♦♥❛❧❛ ♥❛ V ✳ P♦❦❛➸✐t❡✿ ↔❡ ❥❡ Kerf ⊆ Kerg✱

❥❡ g = λf ③❛ ♥❡❦ λ ∈ F ✳

❘❡➨✐t❡✈✳ ❑❡r ❥❡ ❥❡❞r♦ ♥❡♥✐↔❡❧♥❡❣❛ ❧✐♥❡❛r♥❡❣❛ ❢✉♥❦❝✐♦♥❛❧❛ ❤✐♣❡rr❛✈♥✐♥❛✱

♦❜st❛❥❛ t❛❦ 0 = a ∈ V ✱ ❞❛ ❥❡ f(a) = 0✳ ◆❛❥ ❜♦ x ∈ V ✳ P♦t❡♠ ❥❡

x − f(x)a ∈ Kerf ✐♥ ③❛t♦ ❥❡ g(x) = f(x) g(a) ③❛ ✈s❛❦ x ∈ V ✳

f (a)

f (a)

✶✾✳ ◆❛❥ ❜♦st❛ A, B : V → V t❛❦❛ ❡♥❞♦♠♦r✜③♠❛ ❦♦♥↔♥♦ r❛③s❡➸♥❡❣❛ ✈❡❦✲

t♦rs❦❡❣❛ ♣r♦st♦r❛ V ✱ ❞❛ ✈❡❧❥❛ V = ImA ⊕ KerB✳ P♦❦❛➸✐t❡✱ ❞❛ st❛ ③❛❧♦❣✐

✈r❡❞♥♦st✐ ♣r❡s❧✐❦❛✈ BA ✐♥ B ❡♥❛❦✐✳

✾

❘❡➨✐t❡✈✳ ◆❛❥ ❜♦ x ∈ ImB✳ P♦t❡♠ ♦❜st❛❥❛ t❛❦ y ∈ V ✱ ❞❛ ❥❡ x = By✳

P♦ ♣r❡❞♣♦st❛✈❦✐ ❥❡ y = z + z′✱ ❦❥❡r st❛ z ∈ ImA ✐♥ z′ ∈ KerB✳ ❚♦r❡❥ ❥❡

x = Bz✳ ❑❡r ❥❡ z ∈ ImA✱ ❥❡ z = Au ③❛ ♥❡❦ u ∈ V ✳ ■③ t❡❣❛ s❧❡❞✐ ➸❡❧❡♥♦✱

x = BAu✳

✷✵✳ ◆❛❥ ❜♦❞♦ U✱ V ✐♥ W ✈❡❦t♦rs❦✐ ♣r♦st♦r✐ ♥❛❞ ✐st✐♠ ♣♦❧❥❡♠ ✐♥ ♥❛❥ ❜♦st❛

A : U → V t❡r B : V → W ❧✐♥❡❛r♥✐ ♣r❡s❧✐❦❛✈✐✳ P♦❦❛➸✐t❡✿ BA = 0

♥❛t❛♥❦♦ t❡❞❛❥✱ ❦♦ ❥❡ ImA ⊆ KerB✳

❘❡➨✐t❡✈✳ (⇐=) Pr❡❞♣♦st❛✈✐♠♦✱ ❞❛ ❥❡ ImA ⊆ KerB ✐♥ ♥❛❥ ❜♦ u ∈ U✳

P♦t❡♠ ❥❡ Au ∈ ImA✳ ❙❧❡❞✐ BAu = 0 ③❛ ✈s❛❦ u ∈ U✳

(=⇒) ◆❛❥ ❜♦ BA = 0 ✐♥ ♥❛❥ ❜♦ v ∈ ImA✳ P♦t❡♠ ❥❡ v = Au ③❛ ♥❡❦

u ∈ U✳ ❍✐tr♦ ✈✐❞✐♠♦✱ ❞❛ ❥❡ v ∈ KerB✳

✷✶✳ ◆❛❥ ❜♦st❛ D, J : R[x] → R[x] ♣r❡s❧✐❦❛✈✐ ❞❡✜♥✐r❛♥✐ s ♣r❡❞♣✐s♦♠❛ Dp = p′

✐♥ (Jp)(x) = x p(t)dt✳

0

✭❛✮ ❉♦❧♦↔✐t❡ ♣r❡s❧✐❦❛✈✐ DJ ✐♥ JD✳

✭❜✮ ❉♦❧♦↔✐t❡ ♠❛tr✐❦✐ ❧✐♥❡❛r♥✐❤ ♦♣❡r❛t♦r❥❡✈ D ✐♥ J ❣❧❡❞❡ ♥❛ st❛♥❞❛r❞♥♦

❜❛③♦ ♣r♦st♦r❛ r❡❛❧♥✐❤ ♣♦❧✐♥♦♠♦✈ R[x]✳

❘❡➨✐t❡✈✳

✭❛✮ Pr❡s❧✐❦❛✈❛ DJ ❥❡ ✐❞❡♥t✐t❡t❛✱ ♣r❡s❧✐❦❛✈❛ JD ♣❛ ♣r♦❥❡❦t♦r✱ s❛❥ ❥❡

(JD)2 = JD✳

✭❜✮ ▼❛tr✐❦✐ ❧✐♥❡❛r♥✐❤ ♦♣❡r❛t♦r❥❡✈ D ✐♥ J ❣❧❡❞❡ ♥❛ st❛♥❞❛r❞♥♦ ❜❛③♦ ♣r♦s✲

t♦r❛ r❡❛❧♥✐❤ ♣♦❧✐♥♦♠♦✈ R[x] st❛✿













0 1 0 0


0


. . .





0 0 0 0


0


. . .

 0 0 2 0

0

. . . 

 1 0 0 0

0

. . . 









D =  0 0 0 3

0

. . . 

✐♥

 0

1





0 0


0


. . . 





J = 

2

 .

 0 0 0 0

4

. . . 

 0 0

1

0

0

. . . 

 ✳





3



✳ ✳ ✳ ✳

✳ ✳ ✳ ✳

✳ ✳✳ ✳✳ ✳✳ ✳✳✳ · · ·

✳✳ ✳✳ ✳✳ ✳✳ ✳✳✳ ···

✶✵

✷✷✳ ◆❛❥ ❜♦ V ✈❡❦t♦rs❦✐ ♣r♦st♦r✳ P♦❦❛➸✐t❡✿ ↔❡ ③❛ ❧✐♥❡❛r♥♦ ♣r❡s❧✐❦❛✈♦ P : V →

V ✈❡❧❥❛ P 2 = P ✱ ❥❡ V = KerP ⊕ ImP ✳

❘❡➨✐t❡✈✳ ◆❛❥ ❜♦ x ∈ KerP ∩ ImP ✳ P♦t❡♠ ❥❡ P x = 0 ✐♥ x = P y ③❛ ♥❡❦

y ∈ V ✳ ❚♦r❡❥ ❥❡ 0 = P x = P 2y = P y = x✳ Pr❛✈ t❛❦♦ ✈✐❞✐♠♦✱ ❞❛ ❧❛❤❦♦

✈s❛❦ x ∈ V ③❛♣✐➨❡♠♦ ❦♦t x = (x − P x) + P x✱ ❦❥❡r ❥❡ x − P x ∈ KerP ✐♥

P x ∈ ImP ✳

✷✸✳ ◆❛❥ ❜♦ V ✈❡❦t♦rs❦✐ ♣r♦st♦r ✐♥ A : V → V ❧✐♥❡❛r♥✐ ♦♣❡r❛t♦r✳ P♦❦❛➸✐t❡✿

✭❛✮ KerA ⊆ KerA2 ⊆ KerA3 ⊆ . . .

✭❜✮ ◆❛❥ ③❛ ♥❡❦ k ∈ N ✈❡❧❥❛ KerAk = KerAk+1✳ P♦t❡♠ ❥❡ t✉❞✐ KerAk+2 =

KerAk ✭✐♥ ③❛t♦ KerAk = KerAk+j ③❛ ✈s❛❦ j ∈ N✮✳

✭❝✮ ❷❡ ❥❡ ❞✐♠V = n < ∞✱ ♣♦t❡♠ ❥❡ KerAn = KerAn+1 = KerAn+2 =

. . . ❙ ♣r✐♠❡r♦♠ ♣♦❦❛➸✐t❡✱ ❞❛ ❥❡ ❧❛❤❦♦ KerAn−1 = KerAn✳

✭❞✮ ◆❛❥ ❜♦ V ♥❡s❦♦♥↔♥♦ r❛③s❡➸❡♥ ♣r♦st♦r✳ ❙ ♣r✐♠❡r♦♠ ♣♦❦❛➸✐t❡✱ ❞❛

❧❛❤❦♦ ✈❡❧❥❛ KerAn = KerAn+1 ③❛ ✈s❛❦ n ∈ N✳

◆❛♠✐❣✳

✭❛✮ ◆❛❥ ❜♦ x ∈ KerAn✳ P♦t❡♠ ❥❡ An+1x = A(Anx) = 0✱ ❦❛r ♣♦♠❡♥✐✱

❞❛ ❥❡ x ∈ KerAn+1 ③❛ ✈s❛❦ n ∈ N✳

✭❜✮ ●❧❡❞❡ ♥❛ ✭❛✮ ❥❡ KerAk ⊆ KerAk+2✳ Pr❡❞♣♦st❛✈✐♠♦✱ ❞❛ ❥❡ x ∈

KerAk+2✳ P♦t❡♠ ❥❡ Ax ∈ KerAk+1 ✐♥ ③❛t♦ ❥❡ Ax ∈ KerAk✳ ❍✐tr♦

✈✐❞✐♠♦✱ ❞❛ ❥❡ x ∈ KerAk✳

✭❝✮ ❯♣♦➨t❡✈❛❥t❡ ❞✐♠Ai✳ P♦✐➨↔✐t❡ t❛❦♦ r❡❛❧♥♦ n × n ♠❛tr✐❦♦ A✱ ❞❛ ❥❡

An−1 = 0 ✐♥ An = 0✳ ❙ ♣♦♠♦↔❥♦ t❡ ♠❛tr✐❦❡ ♥❛t♦ ❦♦♥str✉✐r❛❥t❡

✐s❦❛♥✐ ♦♣❡r❛t♦r ♥❛ ♥♣r✳ Rn✳

✭❞✮ ◆❛❥ ❜♦ V ✈❡❦t♦rs❦✐ ♣r♦st♦r r❡❛❧♥✐❤ ③❛♣♦r❡❞✐❥ ✐♥ A : V → V ♣r❡s✲

❧✐❦❛✈❛ ❞❡✜♥✐r❛♥❛ s ♣r❡❞♣✐s♦♠ A(x1, x2, x3, . . .) = (x2, x3, . . .)✳

✷✹✳ ◆❛❥ ❜♦ V ✈❡❦t♦rs❦✐ ♣r♦st♦r ✐♥ A : V → V ❧✐♥❡❛r❡♥ ♦♣❡r❛t♦r✳ P♦❦❛➸✐t❡✿

✭❛✮ ImA ⊇ ImA2 ⊇ ImA3 ⊇ . . .

✶✶

✭❜✮ ◆❛❥ ③❛ ♥❡❦ k ∈ N ✈❡❧❥❛ ImAk = ImAk+1✳ P♦t❡♠ ❥❡ t✉❞✐ ImAk+2 =

ImAk ✭✐♥ ③❛t♦ ImAk = ImAk+j ③❛ ✈s❛❦ j ∈ N✮✳

✭❝✮ ❷❡ ❥❡ ❞✐♠V = n < ∞✱ ♣♦t❡♠ ❥❡ ImAn = ImAn+1 = ImAn+2 = . . .

❙ ♣r✐♠❡r♦♠ ♣♦❦❛➸✐t❡✱ ❞❛ ❥❡ ❧❛❤❦♦ ImAn−1 = ImAn✳

◆❛♠✐❣✳ ●❧❡❥t❡ ♥❛♠✐❣ ♣r✐ ♥❛❧♦❣✐ ✷✸✳

✷✺✳ ◆❛❥ ❜♦ M = Mn(R) ❛❧❣❡❜r❛ r❡❛❧♥✐❤ n × n ♠❛tr✐❦✱ n ≥ 2✳ ❩❛ ♣♦❧❥✉❜♥✐

♠❛tr✐❦✐ A, B ∈ M ❞❡✜♥✐r❛❥♠♦ ♣r❡s❧✐❦❛✈♦ L(A,B) : M → M s ♣r❡❞♣✐s♦♠

L(A,B)(X) = AXBT ✳

✭❛✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ L(A,B) ❧✐♥❡❛r❡♥ ♦♣❡r❛t♦r✳ ❑❛❥ ❥❡ L(A,B)L(B,A)❄

✭❜✮ P♦✐➨↔✐t❡ ❜❛③♦ ImL(E11,E12)✳ ❆❧✐ ❥❡ KerL(E11,E12) ❤✐♣❡rr❛✈♥✐♥❛❄

✭❝✮ ◆❛❥ ❜♦





1 1


n = 2 ✐♥ ♥❛❥ ❜♦ A =

✳





2 0


✐✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ B = {E11, E11 − E21, E12 + E21, E22} ❜❛③❛ ♣r♦s✲

t♦r❛ M✳

✐✐✳ ❖♣❡r❛t♦r❥✉ L(A,A) ♣r✐r❡❞✐t❡ ♠❛tr✐❦♦ ❣❧❡❞❡ ♥❛ ❜❛③♦ B✳

◆❛♠✐❣✳

✭❛✮ ❱✐❞✐♠♦✱ ❞❛ ❥❡ L(A,B)L(B,A) = L(AB,BA).

✭❜✮ ❇❛③❛ ♣r♦st♦r❛ ImL(E11,E12) ❥❡ B = {E11}✱ ❞✐♠KerL(E11,E12) = n2−1.

✭❝✮ ✐✳ ❖❜r❛✈♥❛✈❛ ❡♥❛❦♦st✐ αE11+β(E11−E21)+γ(E12+E21)+δE22 =

0 ♥❛s ♣r✐✈❡❞❡ ❞♦ ➸❡❧❡♥❡❣❛✳





1

6





2 1


✐✐✳ ▼❛tr✐❦❛ ♣r✐r❡❥❡♥❛ ♦♣❡r❛t♦r❥✉

 0

−6 0 0 

L(A,A) ❣❧❡❞❡ ♥❛ ❜❛③♦ B ❥❡ 

 .

 2

0

2 0 

4

4





0 0


✷✻✳ ◆❛❥ ❜♦❞♦ P1✱ P2 ✐♥ P3 t❛❦✐ ♣r♦❥❡❦t♦r❥✐ ♥❛ ✈❡❦t♦rs❦❡♠ ♣r♦st♦r✉ X✱ ❞❛ ❥❡

PiPj = −PjPi ③❛ ✈s❡ i = j t❡r P1 + P2 + P3 = I✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡

X = ImP1 ⊕ ImP2 ⊕ ImP3.

✶✷

◆❛♠✐❣✳ ◆❛❥♣r❡❥ ♣♦❦❛➸✐t❡✱ ❞❛ ❥❡ Pij = 0✱ i = j✳ ❷❡ ❥❡ x ∈ X✱ ❥❡

x = Ix = (P1 + P2 + P3)x = P1x + P2x + P3x

✐♥ ③❛t♦ ❥❡ x ∈ ImP1 + ImP2 + ImP3✳

◆❛❥ ❜♦ x ∈ ImP1∩(ImP2⊕ImP3)✳ P♦t❡♠ ❥❡ x = P1x ✐♥ x = P2x2+P3x3

③❛ ♥❡❦❛ x2, x3 ∈ X✳ ■③ t❡❣❛ s❧❡❞✐ P 21x = P1P2x2+P2P3x3. ❚♦r❡❥ ❥❡ P1x = 0

✐♥ ③❛t♦ ❥❡ x = 0✳ P♦❞♦❜♥♦ ♣♦❦❛➸❡♠♦ ➨❡ ♣r❡♦st❛❧❡ ❧❛st♥♦st✐✳

✶✸

P♦❣❧❛✈❥❡ ✷

❇❛♥❛❝❤♦✈✐ ♣r♦st♦r✐

✶✳ ◆❛❥ ❜♦ Y ♥❡♣r❛③♥❛ ♣♦❞♠♥♦➸✐❝❛ ❇❛♥❛❝❤♦✈❡❣❛ ♣r♦st♦r❛ X✱ Z ♣❛ ♥❡♣r❛③♥❛

♣♦❞♠♥♦➸✐❝❛ ❞✉❛❧❛ X∗✳ ◆❛❥ ❜♦

Y ⊥ = {f ∈ X∗ | f(y) = 0, ∀y ∈ Y }

✐♥

Z⊤ = {x ∈ X | f(x) = 0, ∀f ∈ Z}.

P♦❦❛➸✐t❡✿

✭❛✮ Y ⊥ ❥❡ ③❛♣rt ♣♦❞♣r♦st♦r ♣r♦st♦r❛ X∗✳

✭❜✮ Y ⊆ (Y ⊥)⊤.

✭❝✮ ❩❛ ✈s❛❦ ③❛♣rt ♣♦❞♣r♦st♦r Y ⊆ X ❥❡ (Y ⊥)⊤ = Y ✳

❘❡➨✐t❡✈✳

✭❛✮ ◆❛❥ ❜♦ {fn}n∈N t❛❦♦ ③❛♣♦r❡❞❥❡ ✈ Y ⊥✱ ❞❛ ❥❡ limn→∞ fn = f✳ P♦t❡♠

❥❡ f(y) = limn→∞ fn(y) = 0 ③❛ ✈s❛❦ y ∈ Y ✳ ■③ t❡❣❛ s❧❡❞✐✱ ❞❛ ❥❡

f ∈ Y ⊥✳

✭❜✮ ◆❛❥ ❜♦ y ∈ Y ✳ P♦t❡♠ ③❛ ✈s❛❦ f ∈ Y ⊥ ✈❡❧❥❛ f(y) = 0✳ ❚♦r❡❥ ❥❡

y ∈ (Y ⊥)T = {x ∈ X | f(x) = 0, ∀f ∈ Y ⊥}.

✭❝✮ ●❧❡❞❡ ♥❛ ✭❜✮ ♠♦r❛♠♦ ♣♦❦❛③❛t✐✱ ❞❛ ❥❡ (Y ⊥)⊤ ⊆ Y ✳ P♦ ♣♦s❧❡❞✐❝✐

❍❛❤♥ ❇❛♥❛❝❤♦✈❡❣❛ ✐③r❡❦❛ ♦❜st❛❥❛ t❛❦ f ∈ X∗✱ ❞❛ ❥❡ f(Y ) = 0 ✐♥

f (x) = 0 ③❛ ♥❡❦ x ∈ X − Y ✳ ❙❧❡❞✐ x /

∈ (Y ⊥)⊤✳ ❙ t❡♠ s♠♦ ♣♦❦❛③❛❧✐

➸❡❧❡♥♦✳

✶✹

✷✳ ◆❛❥ ❜♦st❛ (V1, . 1) ✐♥ (V2, . 2) ♥♦r♠✐r❛♥❛ ♣r♦st♦r❛✳

✭❛✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ V1 × V2 ♥♦r♠✐r❛♥ ♣r♦st♦r ③ ♥♦r♠♦

(v1, v2) = v1 1 + v2 2.

✭❜✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ V1 × V2 ♥♦r♠✐r❛♥ ♣r♦st♦r ③ ♥♦r♠♦

(v1, v2) = ♠❛①{ v1 1, v2 2}.

◆❛♠✐❣✳ Pr❡✈❡r✐t❡ ✈s❡ ❧❛st♥♦st✐ ③❛ ♥♦r♠✐r❛♥❡ ♣r♦st♦r❡✳

✸✳ ◆❛❥ ❜♦ (Y, . ) ♥♦r♠✐r❛♥ ♣r♦st♦r✱ X ✈❡❦t♦rs❦✐ ♣r♦st♦r ✐♥ A : X → Y

❧✐♥❡❛r❡♥ ♦♣❡r❛t♦r✳ ❩❛ ✈s❛❦ x ∈ X ❞❡✜♥✐r❛♠♦ x 1 = Ax ✳ ❑❛t❡r✐♠

♣♦❣♦❥❡♠ ♠♦r❛ ③❛❞♦➨↔❛t✐ ♦♣❡r❛t♦r A✱ ❞❛ ❜♦ . 1 ♥♦r♠❛ ♥❛ X❄

❘❡➨✐t❡✈✳ ▲✐♥❡❛r♥✐ ♦♣❡r❛t♦r A ❥❡ ✐♥❥❡❦t✐✈❡♥ ♥❛t❛♥❦♦ t❡❞❛❥✱ ❦♦ ❥❡ . 1

♥♦r♠❛✳

✹✳ ◆❛❥ ❜♦st❛ P ✐♥ Q ♣r♦❥❡❦t♦r❥❛ ♥❛ ✈❡❦t♦rs❦❡♠ ♣r♦st♦r✉ X✳

✭❛✮ ❉♦❦❛➸✐t❡✱ ❞❛ s♦ ♥❛s❧❡❞♥❥✐ tr✐❥❡ ♣♦❣♦❥✐ ❡❦✈✐✈❛❧❡♥t♥✐✿

✐✳ P Q = QP = 0,

✐✐✳ P + Q ❥❡ ♣r♦❥❡❦t♦r✱

✐✐✐✳ P Q + QP = 0.

✭❜✮ ◆❛❥ ❜♦ (X, . ) ♥♦r♠✐r❛♥ ♣r♦st♦r ✐♥ ♥❛❥ ♣r♦❥❡❦t♦r❥❛ P t❡r Q ③❛❞♦➨↔❛t❛

♣♦❣♦❥❡♠ ✐③ ✭❛✮✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ s ♣r❡❞♣✐s♦♠ x 1 = P x + Qx

❞❡✜♥✐r❛♥❛ ♥♦r♠❛ ♥❛ X ♥❛t❛♥❦♦ t❡❞❛❥✱ ❦♦ ❥❡ P + Q = I✳

◆❛♠✐❣✳

✭❛✮ ✭✐✐✐ ⇒ ✐✮ ■❞❡♥t✐t❡t♦ P Q + QP = 0 ♣♦♠♥♦➸✐t❡ ③ ♦❜❡❤ str❛♥✐ s P ✐♥

♥❛t♦ ➨❡ s Q✳

✶✺

✭❜✮ (=⇒) Pr❡❞♣♦st❛✈✐♠♦✱ ❞❛ ❥❡ . 1 ♥♦r♠❛ ♥❛ X ✐♥ ♥❛❥ ❜♦ R = P +Q =

I✳ ❑❡r ❥❡ 0 = I − R ♣r♦❥❡❦t♦r✱ ❥❡ KerR = Im(I − R) = 0✳ ■③

t❡❣❛ s❧❡❞✐✱ ❞❛ ♦❜st❛❥❛ t❛❦ 0 = x ∈ X✱ ❞❛ ❥❡ Rx = 0✳ ❚♦r❡❥ ❥❡

0 = P Rx = (P 2 + P Q)x = P x✳ P♦❞♦❜♥♦ ♣♦❦❛➸❡♠♦✱ ❞❛ ❥❡ Qx = 0✳

❙ t❡♠ ♣r✐❞❡♠♦ ✈ ♣r♦t✐s❧♦✈❥❡✱ s❛❥ ❥❡ x 1 = 0 ✐♥ ③❛t♦ x = 0✳

(⇐=) ❷❡ ❥❡ P + Q = I✱ ♥✐ t❡➸❦♦ ♣r❡✈❡r✐t✐✱ ❞❛ ❥❡ s ♣r❡❞♣✐s♦♠ x 1 =

P x + Qx ❞❡✜♥✐r❛♥❛ ♥♦r♠❛ ♥❛ X✳

✺✳ ◆❛❥ ❜♦st❛ (X, . 1) ✐♥ (Y, . 2) ♥♦r♠✐r❛♥❛ ♣r♦st♦r❛✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ X×Y

❇❛♥❛❝❤♦✈ ♣r♦st♦r ③ ♥♦r♠♦ (x, y) = x 1 + y 2 ♥❛t❛♥❦♦ t❡❞❛❥✱ ❦♦ st❛

X ✐♥ Y ❇❛♥❛❝❤♦✈❛ ♣r♦st♦r❛✳

✻✳ ◆❛❥ ❜♦ X ✈❡❦t♦rs❦✐ ♣r♦st♦r ✈s❡❤ ♣♦❧✐♥♦♠♦✈ ③ r❡❛❧♥✐♠✐ ❦♦❡✜❝✐❡♥t✐✳ P♦❦❛➸✐t❡✱

❞❛ ❥❡ s ♣r❡❞♣✐s♦♠

a0 + a1x + a2x2 + . . . + anxn = |a0| + |a1| + |a2| + . . . + |an|

❞❡✜♥✐r❛♥❛ ♥♦r♠❛ ♥❛ X✳

✭❛✮ ❆❧✐ ❥❡ (X, . ) ❇❛♥❛❝❤♦✈ ♣r♦st♦r❄

✭❜✮ ❆❧✐ ❥❡ ♣♦❞♣r♦st♦r ♣♦❧✐♥♦♠♦✈ st♦♣♥❥❡ 2 ③❛♣rt ✈ X❄

◆❛♠✐❣✳

✭❛✮ ❖♣❛③✉❥♠♦ ③❛♣♦r❡❞❥❡ {pn}n∈N✱ ❦❥❡r ❥❡ pn(x) = n 2−kxk. ◆✐ t❡➸❦♦

k=0

♣r❡✈❡r✐t✐✱ ❞❛ ❥❡ t♦ ③❛♣♦r❡❞❥❡ ❈❛✉❝❤②❥❡✈♦✳ Pr❡❞♣♦st❛✈✐♠♦✱ ❞❛ ❥❡

limn→∞ pn = p ∈ X✱ p(x) = b0 + b1x + . . . + bmx✳ P♦t❡♠ ③❛ ✈s❛❦

ǫ > 0 ♦❜st❛❥❛ t❛❦ n0 ∈ N✱ ❞❛ ③❛ ✈s❛❦ n ≥ n0 ✈❡❧❥❛

1

1

1

1

pn − p = |1 − b0| + | − b

− b

+ . . . +

< ǫ.

2

1| + . . . + | 2m

m| + 2m+1

2n

❩❛ ǫ = 1 t♦ ♥❡ ❞r➸✐✳ ■③ t❡❣❛ s❧❡❞✐✱ ❞❛ X ♥✐ ❇❛♥❛❝❤♦✈ ♣r♦st♦r✳

2m+1

✭❜✮ P♦❞♣r♦st♦r ♣♦❧✐♥♦♠♦✈ st♦♣♥❥❡ 2 ❥❡ ③❛♣rt ✈ X✳

✶✻

✼✳ ◆❛❥ ❜♦ X ✈❡❦t♦rs❦✐ ♣r♦st♦r ✈s❡❤ r❡❛❧♥✐❤ ③❛♣♦r❡❞✐❥✱ ❦✐ ✐♠❛❥♦ ❦✈❡↔❥❡♠✉

❦♦♥↔♥♦ ♠♥♦❣♦ ♥❡♥✐↔❡❧♥✐❤ ↔❧❡♥♦✈✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ s ♣r❡❞♣✐s♦♠

{xn}n∈N = max |xn|

n∈N

❞❡✜♥✐r❛♥❛ ♥♦r♠❛ ♥❛ X✳ ❆❧✐ ❥❡ (X, . ) ❇❛♥❛❝❤♦✈ ♣r♦st♦r❄

◆❛♠✐❣✳ Pr♦st♦r X ♥✐ ❇❛♥❛❝❤♦✈✿ ♦♣❛③✉❥t❡ ③❛♣♦r❡❞❥❡ {xn}n∈N✱ ❦❥❡r ❥❡





1 1


1


xn = ( , , . . . ,

, 0, 0, . . .).





2 4


2n


✽✳ ◆❛❥ ❜♦ X ✈❡❦t♦rs❦✐ ♣r♦st♦r ✈s❡❤ ♣♦❧✐♥♦♠♦✈ ③ r❡❛❧♥✐♠✐ ❦♦❡✜❝✐❡♥t✐ ✐♥ ♥❛❥

❜♦ (a0, a1, . . .) ③❛♣♦r❡❞❥❡ ♣♦③✐t✐✈♥✐❤ r❡❛❧♥✐❤ ➨t❡✈✐❧✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ s ♣r❡❞✲

♣✐s♦♠

p = max |anp(n)(0)|

❞❡✜♥✐r❛♥❛ ♥♦r♠❛ ♥❛ X✱ ❦❥❡r ❥❡ p(n) n✲t✐ ♦❞✈♦❞ ♣♦❧✐♥♦♠❛ p✳

✭❛✮ ❆❧✐ ❥❡ (X, . ) ❇❛♥❛❝❤♦✈ ♣r♦st♦r❄

✭❜✮ ❆❧✐ ❥❡ {p ∈ X | p′(0) = p′′(0) = 0} ③❛♣rt ♣♦❞♣r♦st♦r ♣r♦st♦r❛ X❄

◆❛♠✐❣✳

✭❛✮ Pr♦st♦r X ♥✐ ❇❛♥❛❝❤♦✈✿ ♦♣❛③✉❥t❡ ③❛♣♦r❡❞❥❡ {pn}n∈N✱ ❦❥❡r ❥❡

n

xk

pn(x) =

.

k=0 2kk!ak

✭❜✮ Pr♦st♦r ❥❡ ③❛♣rt✳

✾✳ P♦❦❛➸✐t❡✱ ❞❛ st❛ c0 ✐♥ c ③❛♣rt❛ ♣♦❞♣r♦st♦r❛ l∞✳

❘❡➨✐t❡✈✳ ◆❛❥ ❜♦ {xn}n∈N ③❛♣♦r❡❞❥❡ ♣r♦st♦r❛ c0 ✐♥ ♥❛❥ ❜♦ limn→∞ xn = x✳

P♦t❡♠ ③❛ ✈s❛❦ ǫ > 0 ♦❜st❛❥❛ t❛❦ n0 ∈ N✱ ❞❛ ③❛ ✈s❛❦ n ≥ n0 ✈❡❧❥❛

xn − x ∞ = s✉♣|xn − x

k

k| < ǫ✳ ❑❡r ❥❡

|xk| = |xk − xn + x | ≤ x − x

|,

k

nk

n ∞ + |xnk

✶✼

s❧❡❞✐ limn→∞ x = 0✳ ❚♦r❡❥ ❥❡ x ∈ c0✳

◆❛❥ ❜♦ {xn}n∈N ③❛♣♦r❡❞❥❡ ♣r♦st♦r❛ c ✐♥ limn→∞ xn = x✳ ❙ ♣♦♠♦↔❥♦

r❡❧❛❝✐❥❡

|xk − xl| = |xk − xn + x − x + x − x

− x |

k

nk

nl

nl

l| ≤ 2 x − xn ∞ + |xnk

nl

♣♦❦❛➸❡♠♦✱ ❞❛ ❥❡ x ❈❛✉❝❤②❥❡✈♦ ③❛♣♦r❡❞❥❡ ✐♥ ③❛t♦ ❦♦♥✈❡r❣❡♥t♥♦✳ ■③ t❡❣❛

s❧❡❞✐✱ ❞❛ ❥❡ x ∈ c.

✶✵✳ ◆❛❥ ❜♦ X = C[−1, 1] ♣r♦st♦r ③✈❡③♥✐❤ ❢✉♥❦❝✐❥ ♥❛ ③❛♣rt❡♠ ✐♥t❡r✈❛❧✉ [−1, 1]

♦♣r❡♠❧❥❡♥ ③ ♥♦r♠♦ f 1 = 1 |f(x)|dx✳ P♦❦❛➸✐t❡✱ ❞❛ (X, .

−1

1) ♥✐ ♣♦❧♥

♣r♦st♦r✳

◆❛♠✐❣✳ ◆❛❥ ❜♦ {fn}n∈N ③❛♣♦r❡❞❥❡ ❢✉♥❦❝✐❥✱ ❦❥❡r ❥❡ fn : [−1, 1] → R

❢✉♥❦❝✐❥❛ ❞❡✜♥✐r❛♥❛ s ♣r❡❞♣✐s♦♠



 −1 ;

−1 ≤ x ≤ − 1n

fn(x) =

nx ; − 1 < x ≤ 1

.

n

n

 1 ;

1 < x ≤ 1

n

❱✐❞✐♠♦✱ ❞❛ ❥❡

1

1

1

fn − fm 1 =

|fn(x) − fm(x)|dx = | −

|.

−1

n

m

■③ t❡❣❛ s❧❡❞✐✱ ❞❛ ❥❡ {fn}n∈N ❈❛✉❝❤②❥❡✈♦ ③❛♣♦r❡❞❥❡✳

Pr❡❞♣♦st❛✈✐♠♦✱ ❞❛ ❥❡ ③❛♣♦r❡❞❥❡ {fn}n∈N ❦♦♥✈❡r❣❡♥t♥♦✱ limn→∞ fn = f ∈

X✳ ◆❛❥ ❜♦ ǫ > 1 ✳ P♦t❡♠ ❥❡

n

1

1

|fn(x) − f(x)|dx ≥

|1 − f(x)|dx,

−1

ǫ

1

−ǫ

|fn(x) − f(x)|dx ≥

| − 1 − f(x)|dx.

−1

−1

■③ t❡❣❛ s❧❡❞✐✱ ❞❛ ❥❡

−1 ; −1 ≤ x ≤ −ǫ

f (x) =

.

1 ;

ǫ ≤ x ≤ 1

❑❡r f ♥✐ ③✈❡③♥❛ ❢✉♥❦❝✐❥❛✱ s❧❡❞✐ ➸❡❧❡♥♦✳

✶✽

✶✶✳ ◆❛❥ ❜♦st❛ . 1 ✐♥ . 2 ❡❦✈✐✈❛❧❡♥t♥✐ ♥♦r♠✐ ♥❛ ♣r♦st♦r✉ X✳ P♦❦❛➸✐t❡✱ ❞❛

❥❡ (X, . 1) ❇❛♥❛❝❤♦✈ ♣r♦st♦r ♥❛t❛♥❦♦ t❡❞❛❥✱ ❦♦ ❥❡ (X, . 2) ❇❛♥❛❝❤♦✈

♣r♦st♦r✳

✶✷✳ ◆❛ ♣r♦st♦r✉ ③✈❡③♥✐❤ r❡❛❧♥✐❤ ❢✉♥❦❝✐❥ C[0, 1] ✈♣❡❧❥✐♠♦ ♥♦r♠✐ f ∞ = max0≤x≤1 |f(x)|

✐♥ f 1 = max0≤x≤1 |f(x)| + 1 |f(x)|dx✳

0

✭❛✮ ❆❧✐ st❛ ♥♦r♠✐ ❡❦✈✐✈❛❧❡♥t♥✐❄

✭❜✮ ❆❧✐ ❥❡ (C[0, 1], . 1) ❇❛♥❛❝❤♦✈ ♣r♦st♦r❄

◆❛♠✐❣✳

✭❛✮ ◆♦r♠✐ st❛ ❡❦✈✐✈❛❧❡♥t♥✐✳

✭❜✮ ❑❡r ❥❡ (C[0, 1], . ∞) ❇❛♥❛❝❤♦✈ ♣r♦st♦r✱ ♣♦ ♥❛❧♦❣✐ ✶✶ ✈❡❧❥❛✱ ❞❛ ❥❡

t✉❞✐ (C[0, 1], . 1) ❇❛♥❛❝❤♦✈ ♣r♦st♦r✳

✶✸✳ ◆❛ ♣r♦st♦r✉ X = C[0, 1] ✈♣❡❧❥✐♠♦ ♥♦r♠❡ f ∞ = max0≤x≤1 |f(x)|✱ f 1 =

1

( 1 |f(x)|2dx)2 ✐♥ f

0

2 =

f 1 + f ∞✳

✭❛✮ ❆❧✐ st❛ ♥♦r♠✐ . ∞ ✐♥ . 1 ❡❦✈✐✈❛❧❡♥t♥✐❄

✭❜✮ ❩❛ ❦❛t❡r♦ ♥♦r♠♦ ❥❡ X ❇❛♥❛❝❤♦✈ ♣r♦st♦r❄

◆❛♠✐❣✳

✭❛✮ ◆♦r♠✐ ♥✐st❛ ❡❦✈✐✈❛❧❡♥t♥✐✱ s❛❥ (X, . 1) ♥✐ ❇❛♥❛❝❤♦✈ ♣r♦st♦r ✭✉♣♦➨t❡✲

✈❛❥t❡ ♥❛❧♦❣♦ ✶✶✮✳

✭❜✮ Pr♦st♦r❛ (X, . ∞) ✐♥ (X, . 2) st❛ ❇❛♥❛❝❤♦✈❛ ♣r♦st♦r❛✳

✶✹✳ ◆❛❥ ❜♦ (X, . ) ♥♦r♠✐r❛♥ ♣r♦st♦r✳ ❱ ✈❡❦t♦rs❦✐ ♣r♦st♦r W = X × X

✈♣❡❧❥✐♠♦ ♥♦r♠♦ (x, y) = x + y .

✭❛✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ s ♣r❡❞♣✐s♦♠ (x, y) 1 = (x−y, y) ❞❡✜♥✐r❛♥❛ ♥♦r♠❛

♥❛ W ✳

✭❜✮ ◆❛❥ ❜♦ (W, . ) ❇❛♥❛❝❤♦✈ ♣r♦st♦r✳ ❆❧✐ ❥❡ (W, . 1) ❇❛♥❛❝❤♦✈ ♣r♦s✲

t♦r❄

✶✾

◆❛♠✐❣✳ ✭❜✮ Pr♦st♦r (W, . 1) ❥❡ ❇❛♥❛❝❤♦✈✿ ♣♦❦❛➸✐t❡✱ ❞❛ ❥❡

1 (x,y) ≤ (x,y)

2

1 ≤ 2 (x, y)

✐♥ ✉♣♦➨t❡✈❛❥t❡ ♥❛❧♦❣♦ ✶✶✳

✶✺✳ ◆❛❥ ❜♦ (X, . ) ♥♦r♠✐r❛♥ ♣r♦st♦r ✐♥ ♥❛❥ ❜♦ A : X → X ❧✐♥❡❛r❡♥ ♦♣❡r❛t♦r✳

P♦❦❛➸✐t❡✿

✭❛✮ ❙ ♣r❡❞♣✐s♦♠ x 1 = x − Ax + Ax ❥❡ ❞❡✜♥✐r❛♥❛ ♥♦r♠❛ ♥❛ X✳

✭❜✮ ❷❡ ❥❡ (X, . ) ❇❛♥❛❝❤♦✈ ♣r♦st♦r✱ ❥❡ ✈s❛❦♦ ❈❛✉❝❤②❥❡✈♦ ③❛♣♦r❡❞❥❡ ✈

(X, . 1) ❦♦♥✈❡r❣❡♥t♥♦ ✈ (X, . )✳

✭❝✮ ❷❡ ❥❡ A ♦♠❡❥❡♥ ♦♣❡r❛t♦r✱ ❥❡ (X, . ) ❇❛♥❛❝❤♦✈ ♣r♦st♦r ♥❛t❛♥❦♦

t❡❞❛❥✱ ❦♦ ❥❡ (X, . 1) ❇❛♥❛❝❤♦✈ ♣r♦st♦r✳

◆❛♠✐❣✳

✭❜✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ x ≤ x 1✱ ✐③ ↔❡s❛r s❧❡❞✐ ➸❡❧❡♥♦✳

✭❝✮ P♦❦❛➸✐t❡✱ ❞❛ st❛ ♥♦r♠✐ . ✐♥ . 1 ❡❦✈✐✈❛❧❡♥t♥✐ ✐♥ ✉♣♦➨t❡✈❛❥t❡ ♥❛❧♦❣♦

✶✶✳

✶✻✳ ◆❛❥ ❜♦st❛ P ✐♥ Q t❛❦❛ ♣r♦❥❡❦t♦r❥❛ ♥❛ ♥♦r♠✐r❛♥❡♠ ♣r♦st♦r✉ (X, . )✱ ❞❛

❥❡ t✉❞✐ P + Q ♣r♦❥❡❦t♦r✳

✭❛✮ ❉♦❦❛➸✐t❡✱ ❞❛ ❥❡ s ♣r❡❞♣✐s♦♠ x 1 = ♠❛① { P x , Qx } ❞❡✜♥✐r❛♥❛

♥♦r♠❛ ♥❛ X ♥❛t❛♥❦♦ t❡❞❛❥✱ ❦♦ ❥❡ P = I − Q✳

✭❜✮ ◆❛❥ ❜♦ P ∈ B(X) ✐♥ ♥❛❥ ❜♦ (X, . ) ❇❛♥❛❝❤♦✈ ♣r♦st♦r✳ ❆❧✐ ❥❡ t❡❞❛❥

♥♦r♠✐r❛♥ ♣r♦st♦r (X, . 1) ❇❛♥❛❝❤♦✈❄

◆❛♠✐❣✳ ✭❜✮ ❑❡r st❛ ♥♦r♠✐ . ✐♥ . 1 ❡❦✈✐✈❛❧❡♥t♥✐✱ s❧❡❞✐✱ ❞❛ ❥❡ (X, . 1)

❇❛♥❛❝❤♦✈ ♣r♦st♦r✳

✷✵

✶✼✳ ◆❛❥ ❜♦ (X, . ) ❇❛♥❛❝❤♦✈ ♣r♦st♦r ✐♥ ♥❛❥ ❜♦ G ♣♦❞♠♥♦➸✐❝❛ B(X)✳ ❉❡♥✲

✐♠♦✱ ❞❛ ❥❡ G t❛❦❛ ❣r✉♣❛ ✭③❛ ♦♣❡r❛❝✐❥♦ ♠♥♦➸❡♥❥❛ ♦♣❡r❛t♦r❥❡✈✮✱ ❞❛ ③❛ ✈s❛❦

T ∈ G ✈❡❧❥❛ T ≤ 1✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ s ♣r❡❞♣✐s♦♠ x 0 = s✉♣T∈G T x

❞❡✜♥✐r❛♥❛ ♥♦r♠❛ ♥❛ X✱ ❦✐ ❥❡ ❡❦✈✐✈❛❧❡♥t♥❛ ♥♦r♠✐ . ✐♥ ✈ ❦❛t❡r✐ s♦ ✈s✐

♦♣❡r❛t♦r❥✐ T ∈ G ✐③♦♠❡tr✐❥❡✳

❘❡➨✐t❡✈✳ ❑❡r ❥❡ T ≤ 1 ③❛ ✈s❛❦ T ∈ G✱ ❥❡ x 0 ≤ x ③❛ ✈s❛❦ x ∈ X✳

P♦ ❞r✉❣✐ str❛♥✐ ♣❛ ❥❡ x = Ix ≤ x 0✳ ❚♦r❡❥ st❛ ♥♦r♠✐ . ✐♥ . 0

❡❦✈✐✈❛❧❡♥t♥✐✳

❱③❡♠✐♠♦ ♣♦❧❥✉❜❡♥ S ∈ G✳ P♦t❡♠ ❥❡

Sx 0 = s✉♣

T Sx = s✉♣

T x = x

T ∈G

T ∈G

0

③❛ ✈s❛❦ x ∈ X✱ ❦❛r ♣♦♠❡♥✐✱ ❞❛ ❥❡ S ✐③♦♠❡tr✐❥❛ ✈ ♥♦r♠✐ . 0✳

✶✽✳ ◆❛❥ ❜♦ a ♣♦③✐t✐✈♥♦ r❡❛❧♥♦ ➨t❡✈✐❧♦✳ ❩❛ ✈s❛❦ n ∈ N ❞❡✜♥✐r❛❥♠♦ ♦♣❡r❛t♦r

An : l∞ → l∞ s ♣r❡❞♣✐s♦♠

An(x1, x2, . . .)

1

=

(x

a

1, x2, . . . , xn, −xn+1, xn+2, . . .).

✭❛✮ ❑❛❦➨❡♥ ♠♦r❛ ❜✐t✐ a✱ ❞❛ ❜♦ ③❛♣♦r❡❞❥❡ (An, A2, A3, . . .) ❦♦♥✈❡r❣❡♥t♥♦

n

n

♦③✐r♦♠❛ ❜♦ ✐♠❡❧♦ ❦❛❦♦ ❦♦♥✈❡r❣❡♥t♥♦ ♣♦❞③❛♣♦r❡❞❥❡❄

✭❜✮ ❆❧✐ ❥❡ ③❛♣♦r❡❞❥❡ {An}n∈N ❦♦♥✈❡r❣❡♥t♥♦ ✈ ♦♣❡r❛t♦rs❦✐ ♥♦r♠✐❄

✭❝✮ ◆❛❥ ❜♦ x = (x1, x2, x3 . . .) ③❛♣♦r❡❞❥❡ ③ ❧✐♠✐t♦ 0✳ ❆❧✐ ❥❡ ③❛♣♦r❡❞❥❡

{Anx}n∈N ❦♦♥✈❡r❣❡♥t♥♦❄

◆❛♠✐❣✳

✭❛✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡

1

1

1

1

(An, A2 , A3 , A4 , . . .) = (A

I,

A

I,

A

n

n

n

n, a2 a2 n, a4 a4 n, . . .).

❚♦ ③❛♣♦r❡❞❥❡ ❜♦ ❦♦♥✈❡r❣❡♥t♥♦✱ ❦♦ ❜♦ a > 1✱ ✐♥ ❜♦ ✐♠❡❧♦ ❦♦♥✈❡r✲

❣❡♥t♥♦ ♣♦❞③❛♣♦r❡❞❥❡✱ ↔❡ ❜♦ a = 1✳

✭❜✮ ❑❡r ③❛ n = m ✈❡❧❥❛ An − Am ∞ = 2✱ ③❛♣♦r❡❞❥❡ {An}n∈N ♥✐ ❦♦♥✲

✈❡r❣❡♥t♥♦✳

✷✶

✭❝✮ ❩❛♣♦r❡❞❥❡ ❥❡ ❦♦♥✈❡r❣❡♥t♥♦✳

✶✾✳ ◆❛❥ ❜♦ ♣r♦st♦r X = C[−1, 1] ♦♣r❡♠❧❥❡♥ ③ ♥♦r♠♦ f ∞ = max−1≤x≤1 |f(x)|

✐♥ ♥❛❥ ❜♦ F : (X, . ∞) → C ♣r❡s❧✐❦❛✈❛ ❞❡✜♥✐r❛♥❛ s ♣r❡❞♣✐s♦♠

1

F (f ) =

(f (t) − f(−t))dt.

0

P♦❦❛➸✐t❡✱ ❞❛ ❥❡ F ♦♠❡❥❡♥ ❧✐♥❡❛r❡♥ ❢✉♥❦❝✐♦♥❛❧ ✐♥ ✐③r❛↔✉♥❛❥t❡ ♥❥❡❣♦✈♦

♥♦r♠♦✳

◆❛♠✐❣✳ ❖♣❛③✉❥t❡ ③❛♣♦r❡❞❥❡ ❢✉♥❦❝✐❥✱ ❦✐ ❥❡ ❞❡✜♥✐r❛♥♦ ✈ ♥❛❧♦❣✐ ✶✵ ✐♥ ✉♣✲

♦➨t❡✈❛❥t❡✱ ❞❛ ❥❡

1

1

1

|F (fn)| = |

(fn(x) − fn(−x)dx| = 2|

fn(x)dx| = 2 −

0

0

n

◆♦r♠❛ ♦♣❡r❛t♦r❥❛ F ❥❡ ✷✳

✷✵✳ ◆❛❥ ❜♦ ♣r♦st♦r X = C[0, 1] ♦♣r❡♠❧❥❡♥ ③ ♥♦r♠♦ f ∞ = max0≤x≤1 |f(x)|✳

P♦❦❛➸✐t❡✱ ❞❛ ❥❡ s ♣r❡❞♣✐s♦♠ ϕ(f) = f(1)−f(0) ❞❡✜♥✐r❛♥ ♦♠❡❥❡♥ ❧✐♥❡❛r❡♥

❢✉♥❦❝✐♦♥❛❧ ♥❛ X ✐♥ ✐③r❛↔✉♥❛❥t❡ ♥❥❡❣♦✈♦ ♥♦r♠♦✳

❘❡➨✐t❡✈✳ ◆♦r♠❛ ♦♣❡r❛t♦r❥❛ ϕ ❥❡ 2✳

✷✶✳ ◆❛❥ ❜♦ A : c → c0 ♣r❡s❧✐❦❛✈❛ ❞❡✜♥✐r❛♥❛ s ♣r❡❞♣✐s♦♠

x

A({x

n

n}n∈N) = {

}

n n∈N.

P♦❦❛➸✐t❡✱ ❞❛ ❥❡ A ♦♠❡❥❡♥ ❧✐♥❡❛r❡♥ ❢✉♥❦❝✐♦♥❛❧ ✐♥ ✐③r❛↔✉♥❛❥t❡ ♥❥❡❣♦✈♦

♥♦r♠♦✳

❘❡➨✐t❡✈✳ ◆♦r♠❛ ♦♣❡r❛t♦r❥❛ A ❥❡ 1✳

✷✷

✷✷✳ ◆❛❥ ❜♦ X ♥♦r♠✐r❛♥ ♣r♦st♦r✱ dim X ≥ 2✱ a ∈ X✱ f ∈ X∗ ✐♥ ♥❛❥ ❜♦

A : X → X ♦♣❡r❛t♦r ❞❡✜♥✐r❛♥ s ♣r❡❞♣✐s♦♠ Ax = f(x)a✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡

A ♦♠❡❥❡♥ ❧✐♥❡❛r❡♥ ♦♣❡r❛t♦r ✐♥ ✐③r❛↔✉♥❛❥t❡ ♥❥❡❣♦✈♦ ♥♦r♠♦✳ ❆❧✐ ❥❡ ♦♣❡r❛t♦r

A ✐♥❥❡❦t✐✈❡♥❄ ❆❧✐ ❥❡ ♦♣❡r❛t♦r A s✉r❥❡❦t✐✈❡♥❄

❘❡➨✐t❡✈✳ ◆♦r♠❛ ♦♣❡r❛t♦r❥❛ A ❥❡ f a ✳ ❖♣❡r❛t♦r A ♥✐ ♥✐t✐ ✐♥❥❡❦t✐✈❡♥

♥✐t✐ s✉r❥❡❦t✐✈❡♥✳

✷✸✳ ◆❛❥ ❜♦ ♣r♦st♦r X = C[0, b] ♦♣r❡♠❧❥❡♥ ③ ♥♦r♠♦ f ∞ = max0≤x≤b |f(x)| ✐♥

♥❛❥ ❜♦ A : X → X ♦♣❡r❛t♦r ❞❡✜♥✐r❛♥ s ♣r❡❞♣✐s♦♠ (Af)(x) = f(x) cos x✳

✭❛✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ A ∈ B(X) ✐♥ ✐③r❛↔✉♥❛❥t❡ ♥❥❡❣♦✈♦ ♥♦r♠♦✳

✭❜✮ ❑❛❦➨♥♦ ♠♦r❛ ❜✐t✐ ➨t❡✈✐❧♦ b✱ ❞❛ ❜♦ ♦♣❡r❛t♦r A ♦❜r♥❧❥✐✈❄ ❑❛t❡r✐ ♦♣✲

❡r❛t♦r ❥❡ t❡❞❛❥ ♥❥❡❣♦✈ ✐♥✈❡r③❄

✭❝✮ ■③r❛↔✉♥❛❥t❡ s♣❡❦t❡r ♦♣❡r❛t♦r❥❛ A✳

❘❡➨✐t❡✈✳

✭❛✮ Af ∞ ≤ f ∞✱ A = 1✳

✭❜✮ b < π✱ (A−1f)(x) = f(x)(cos x)−1✳

2

✭❝✮ σ(A) = {cos x | x ∈ [0, b]}✳

✷✹✳ ◆❛❥ ❜♦ ♣r♦st♦r X = C[1, b] ♦♣r❡♠❧❥❡♥ ③ ♥♦r♠♦ f

2

∞ = max 1 ≤x≤b |f (x)|

2

✐♥ ♥❛❥ ❜♦ A : X → X ♦♣❡r❛t♦r ❞❡✜♥✐r❛♥ s ♣r❡❞♣✐s♦♠ (Af)(x) = f(x) ln x✳

✭❛✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ A ∈ B(X) ✐♥ ✐③r❛↔✉♥❛❥t❡ ♥❥❡❣♦✈♦ ♥♦r♠♦✳

✭❜✮ ❑❛❦➨♥♦ ♠♦r❛ ❜✐t✐ ➨t❡✈✐❧♦ b✱ ❞❛ ❜♦ ♦♣❡r❛t♦r A ♦❜r♥❧❥✐✈❄ ❑❛t❡r✐ ♦♣❡✲

r❛t♦r ❥❡ t❡❞❛❥ ♥❥❡❣♦✈ ✐♥✈❡r③❄

✭❝✮ ■③r❛↔✉♥❛❥t❡ s♣❡❦t❡r ♦♣❡r❛t♦r❥❛ A✳

❘❡➨✐t❡✈✳

✭❛✮ A = max1≤x≤b |ln x|.

2

✭❜✮ b < 1✱ (A−1f)(x) = f(x)(ln x)−1✳

✷✸

✭❝✮ σ(A) = {ln x | x ∈ [1, b]}.

2

✷✺✳ ◆❛ ♣r♦st♦r✉ X = C[0, 1] ✈♣❡❧❥✐♠♦ ♥♦r♠♦

1

f = max |f(x)| +

|f(x)|dx.

0≤x≤1

0

◆❛❥ ❜♦ A : X → X ♦♣❡r❛t♦r ❞❡✜♥✐r❛♥ s ♣r❡❞♣✐s♦♠ (Af)(x) = xf(x)✳

✭❛✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ (X, . ) ❇❛♥❛❝❤♦✈ ♣r♦st♦r✳

✭❜✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ A ∈ B(X)✳

✭❝✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡

∞

An ❦♦♥✈❡r❣❡♥t♥❛ ✈rst❛✳

n=0 n!

✭❞✮ ❆❧✐ ✐♠❛ ♦♣❡r❛t♦r ❦❛❦♦ ❧❛st♥♦ ✈r❡❞♥♦st❄

◆❛♠✐❣✳

✭❛✮ ❯♣♦➨t❡✈❛❥t❡ ❡❦✈✐✈❛❧❡♥t♥♦st ♥♦r♠✿

f ∞ = max |f(x)| ≤ f ≤ 2 f ∞.

0≤x≤1

✭❜✮ A ≤ 2✳

✭❝✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ An ≤ 2 ✐♥ ✉♣♦➨t❡✈❛❥t❡

∞

An

2 .

n=0

≤

∞

n!

n=0 n!

✭❞✮ ❖♣❡r❛t♦r ♥✐♠❛ ❧❛st♥✐❤ ✈r❡❞♥♦st✐✳

✷✻✳ ◆❛ ♣r♦st♦r✉ X = C[0, 1] ✈♣❡❧❥✐♠♦ ♥♦r♠♦

1

1

f = 2 max |f(x)| + (

|f(x)|2dx)2 .

0≤x≤1

0

◆❛❥ ❜♦ A : X → X ♦♣❡r❛t♦r ❞❡✜♥✐r❛♥ s ♣r❡❞♣✐s♦♠ (Af)(x) = e−xf(x)✳

✭❛✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ (X, . ) ❇❛♥❛❝❤♦✈ ♣r♦st♦r✳

✭❜✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ A ∈ B(X)✳

✭❝✮ ❆❧✐ ✈rst❛

∞

An ❦♦♥✈❡r❣✐r❛❄

n=0 n!

✭❞✮ ❩❛♣✐➨✐t❡ A−1✳ ❆❧✐ ✐♠❛ ♦♣❡r❛t♦r A ❦❛❦♦ ❧❛st♥♦ ✈r❡❞♥♦st❄

✷✹

❘❡➨✐t❡✈✳

✭❛✮ ❯♣♦➨t❡✈❛❥t❡ ❡❦✈✐✈❛❧❡♥t♥♦st ♥♦r♠✿

f ∞ = max |f(x)| ≤ f ≤ 3 f ∞.

0≤x≤1

✭❜✮ A ≤ 3✳

✭❝✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ An ≤ 3 ✐♥ ✉♣♦➨t❡✈❛❥t❡

∞

An

3 .

n=0

≤

∞

n!

n=0 n!

✭❞✮ ❖♣❡r❛t♦r ♥✐♠❛ ❧❛st♥✐❤ ✈r❡❞♥♦st✐✳

✷✼✳ ◆❛❥ ❜♦ ♣r♦st♦r X = C[0, 1] ♦♣r❡♠❧❥❡♥ ③ ♥♦r♠♦ f ∞ = max0≤x≤1 |f(x)| ✐♥

♥❛❥ ❜♦ V : X → X ♦♣❡r❛t♦r ❞❡✜♥✐r❛♥ s ♣r❡❞♣✐s♦♠ (V f)(x) = x f(t)dt✳

0

P♦❦❛➸✐t❡✱ ❞❛ ❥❡ V ∈ B(X) ❦✈❛③✐♥✐❧♣♦t❡♥t❡♥ ♦♣❡r❛t♦r✳

◆❛♠✐❣✳ ◆❛❥♣r❡❥ ♣♦❦❛➸✐t❡✱ ❞❛ ❥❡ |(V nf)(x)| ≤ xn f ✱ ✐③ ↔❡s❛r s❧❡❞✐

n!

V n ≤ 1 ✳ ❚♦r❡❥ ❥❡

n!





1 1


r(V ) = lim V n 1n ≤ lim ( )n = 0.

n→∞

n→∞ n!

✷✽✳ ◆❛❥ ❜♦ ♣r♦st♦r X = C[0, 1] ♦♣r❡♠❧❥❡♥ ③ ♥♦r♠♦ f ∞ = max0≤x≤1 |f(x)|

✐♥ ♥❛❥ ❜♦st❛ M, V : X → X ♣r❡s❧✐❦❛✈✐ ❞❡✜♥✐r❛♥✐ s ♣r❡❞♣✐s♦♠❛ (Mf)(x) =

xf (x) t❡r (V f)(x) = x f(t)dt✳ P♦❦❛➸✐t❡✱ ❞❛ s♦ M, V, MV −V M ∈ B(X)✱

0

✐③r❛↔✉♥❛❥t❡ ♥❥✐❤♦✈❡ ♥♦r♠❡ t❡r ♣♦❦❛➸✐t❡✱ ❞❛ ❥❡ MV −V M ❦✈❛③✐♥✐❧♣♦t❡♥t❡♥

♦♣❡r❛t♦r✳

◆❛♠✐❣✳ ❱✐❞✐♠♦✱ ❞❛ ❥❡

x

x

((M V − V M)f)(x) = x

f (t)dt −

tf (t)dt = (V 2f )(x).

0

0

❚♦r❡❥ ❥❡ MV − V M = V 2✳ P♦ ♣r❡❥➨♥❥✐ ♥❛❧♦❣✐ ❥❡ σ(V 2) = σ(V )2 = 0✳

✷✺

✷✾✳ ◆❛❥ ❜♦ ♣r♦st♦r X = C[0, 1] ♦♣r❡♠❧❥❡♥ ③ ♥♦r♠♦ f ∞ = max0≤x≤1 |f(x)|

✐♥ ♥❛❥ ❜♦ P ♣♦❞♣r♦st♦r ✈s❡❤ ♣♦❧✐♥♦♠♦✈✳ P♦❦❛➸✐t❡✱ ❞❛ st❛ s ♣r❡❞♣✐s♦♠❛

f (a0 + a1x + . . . + anxn) = a0,

g(a0 + a1x + . . . + anxn) = a0 + a1 + . . . + an

❞❡✜♥✐r❛♥❛ ♦♠❡❥❡♥❛ ❧✐♥❡❛r♥❛ ❢✉♥❦❝✐♦♥❛❧❛ ♥❛ P ✐♥ ✐③r❛↔✉♥❛❥t❡ ♥❥✉♥✐ ♥♦r♠✐✳

P♦✐➨↔✐t❡ r❛③➨✐r✐t✈✐ ❢✉♥❦❝✐♦♥❛❧♦✈ f ✐♥ g ♥❛ ♣r♦st♦r X✱ ❦✐ ✐♠❛t❛ ✐st✐ ♥♦r♠✐

❦♦t f ♦③✐r♦♠❛ g✳

❘❡➨✐t❡✈✳ ◆♦r♠❛ ❢✉♥❝✐♦♥❛❧♦✈ f ✐♥ g ❥❡ 1✳ ❘❛③➨✐r✐t❡✈ ❢✉♥❦❝✐♦♥❛❧❛ f ❥❡ ♥♣r✳

F (ϕ) = ϕ(0)✱ ϕ ∈ X✱ ❢✉♥❦❝✐♦♥❛❧❛ g ♣❛ G(ϕ) = ϕ(1)✱ ϕ ∈ X✳

✸✵✳ ◆❛❥ ❜♦ ♣r♦st♦r ③✈❡③♥♦ ♦❞✈❡❞❧❥✐✈✐❤ r❡❛❧♥✐❤ ❢✉♥❦❝✐❥ X = C1[0, 1] ♦♣r❡♠❧❥❡♥

③ ♥♦r♠♦ f = supx∈[0,1] |f(x)| + supx∈[0,1] |f′(x)| ✐♥ ♥❛❥ ❜♦ P ♣♦❞♣r♦st♦r

✈s❡❤ ♣♦❧✐♥♦♠♦✈✳ P♦❦❛➸✐t❡✱ ❞❛ st❛ s ♣r❡❞♣✐s♦♠❛

ϕ(a0 + a1x + . . . + anxn) = a0 + 5−1a1 + 5−2a2 + . . . + 5−nan,

ψ(a0 + a1x + . . . + anxn) = 3a0 + 6a1 + 9a2 + 12a3 + . . . + 3(n + 1)an

❞❡✜♥✐r❛♥❛ ♦♠❡❥❡♥❛ ❧✐♥❡❛r♥❛ ❢✉♥❦❝✐♦♥❛❧❛ ♥❛ P ✐♥ ✐③r❛↔✉♥❛❥t❡ ♥❥✉♥✐ ♥♦r♠✐✳

P♦✐➨↔✐t❡ r❛③➨✐r✐t✈✐ ❢✉♥❦❝✐♦♥❛❧♦✈ ϕ ✐♥ ψ ♥❛ ❝❡❧ ♣r♦st♦r X✱ ❦✐ ✐♠❛t❛ ✐st✐

♥♦r♠✐ ❦♦t ϕ ♦③✐r♦♠❛ ψ✳

❘❡➨✐t❡✈✳ ❖♣❛③✐♠♦✱ ❞❛ ❥❡ ϕ(p) = p(1)✱ ψ(p) = 3(p(1) + p′(1)) ✐♥ ✐③r❛↔✉✲

5

♥❛♠♦✱ ❞❛ ❥❡ ϕ = 1 t❡r ψ = 3✳ ❘❛③➨✐r✐t❡✈ ❢✉♥❦❝✐♦♥❛❧❛ ϕ ❥❡ ♥❛

♣r✐♠❡r ϕ1(f) = f(1)✱ f ∈ X✱ r❛③➨✐r✐t❡✈ ❢✉♥❦❝✐♦♥❛❧❛ ψ ♣❛ ψ

5

1(f ) =

3(f (x) + f ′(x))✱ f ∈ X✳

✸✶✳ ◆❛❥ ❜♦ X = C[0, 1] ♣r♦st♦r ♦♣r❡♠❧❥❡♥ ③ ♥♦r♠♦ f ∞ = max0≤x≤1 |f(x)|✳

❩ L ♦③♥❛↔✐♠♦ ♠♥♦➸✐❝♦ ✈s❡❤ ❧✐❤✐❤ ❢✉♥❦❝✐❥ ♣r♦st♦r❛ X ✐♥ ③ S ♠♥♦➸✐❝♦ ✈s❡❤

s♦❞✐❤ ❢✉♥❦❝✐❥ ♣r♦st♦r❛ X✳

✭❛✮ P♦❦❛➸✐t❡✱ ❞❛ st❛ L ✐♥ S ♣♦❞♣r♦st♦r❛ X ✐♥ ❞❛ ❥❡ L ⊕ S = X.

✭❜✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ s ♣r❡❞♣✐s♦♠

1

ϕ(f ) =

f (x)dx

− 12

✷✻

❞❡✜♥✐r❛♥ ♦♠❡❥❡♥ ❧✐♥❡❛r❡♥ ❢✉♥❦❝✐♦♥❛❧ ♥❛ L ✐♥ ✐③r❛↔✉♥❛❥t❡ ♥❥❡❣♦✈♦

♥♦r♠♦✳

✭❝✮ P♦✐➨↔✐t❡ t❛❦♦ r❛③➨✐r✐t❡✈ φ ❢✉♥❦❝✐♦♥❛❧❛ ϕ ♥❛ X✱ ❞❛ ❥❡ φ = ϕ ✳

◆❛♠✐❣✳ ◆♦r♠❛ ❢✉♥❝✐♦♥❛❧❛ ϕ ❥❡ 1✳ ❘❛③➨✐r✐t❡✈ ❢✉♥❦❝✐♦♥❛❧❛ ϕ ❥❡ ♥❛ ♣r✐♠❡r

2

φ(f ) = 11 f (x)dx✱ f ∈ X✳

2

✸✷✳ ◆❛❥ ❜♦ ♣r♦st♦r X = C[0, 1] ♦♣r❡♠❧❥❡♥ ③ ♥♦r♠♦ f ∞ = max0≤x≤1 |f(x)|

✐♥ ♥❛❥ ❜♦ Y = {f ∈ X | f(0) = 0}✳

✭❛✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ Y ③❛♣rt ♣♦❞♣r♦st♦r ♣r♦st♦r❛ X✳

✭❜✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ s ♣r❡❞♣✐s♦♠ ϕ(f) = f(1) − f(0) ❞❡✜♥✐r❛♥ ♦♠❡❥❡♥

2

❧✐♥❡❛r❡♥ ❢✉♥❦❝✐♦♥❛❧ ♥❛ X✳ ■③r❛↔✉♥❛❥t❡ ♥❥❡❣♦✈♦ ♥♦r♠♦ ✐♥ ♥♦r♠♦

♥❥❡❣♦✈❡ ③♦➸✐t✈❡ ♥❛ Y ✳

✭❝✮ P♦✐➨↔✐t❡ ❦❛❦ t❛❦ ❧✐♥❡❛r♥✐ ❢✉♥❦❝✐♦♥❛❧ ♥❛ X✱ ❦✐ s❡ ♥❛ Y ✉❥❡♠❛ s ϕ ✐♥

✐♠❛ ✐st♦ ♥♦r♠♦ ❦♦t ③♦➸✐t❡✈ ϕ ♥❛ Y ✳

◆❛♠✐❣✳ ◆♦r♠❛ ❢✉♥❝✐♦♥❛❧❛ ϕ ❥❡ 1✳ ❘❛③➨✐r✐t❡✈ ❢✉♥❦❝✐♦♥❛❧❛ ϕ ❥❡ ♥❛ ♣r✐♠❡r

φ(f ) = f (1 )✱ f ∈ X✳

2

✸✸✳ ◆❛❥ ❜♦ (X, . ) ♥♦r♠✐r❛♥ ♣r♦st♦r✱ a1, a2, . . . , an ∈ C ✐♥ x1, x2, . . . , xn

❧✐♥❡❛r♥♦ ♥❡♦❞✈✐s♥✐ ✈❡❦t♦r❥✐ ♣r♦st♦r❛ X✳ P♦❦❛➸✐t❡✱ ❞❛ st❛ ♥❛s❧❡❞♥❥✐ ❞✈❡

tr❞✐t✈✐ ❡❦✈✐✈❛❧❡♥t♥✐✿

✭❛✮ ♦❜st❛❥❛ t❛❦ f ∈ X∗✱ ❞❛ ❥❡ f ≤ 1 ✐♥ f(xi) = ai✱ i = 1, . . . , n✱

✭❜✮ ③❛ ♣♦❧❥✉❜♥❡ b1, b2, . . . , bn ∈ C ❥❡

|b1a1 + b2a2 + . . . + bnan| ≤ b1x1 + b2x2 + . . . + bnxn .

❘❡➨✐t❡✈✳ ✭✭❛✮ ⇒ ✭❜✮✮ ❖♣❛③✐♠♦✱ ❞❛ ❥❡

|b1a1 + b2a2 + . . . + bnan| = |f(b1x1 + b2x2 + . . . + bnxn)|

≤

f

b1x1 + b2x2 + . . . + bnxn

≤

b1x1 + b2x2 + . . . + bnxn .

✷✼

✭✭❜✮ ⇒ ✭❛✮✮ ◆❛❥ ❜♦ Y = L{x1, x2, . . . , xn}✳ ❉❡✜♥✐r❛❥♠♦ f1 : Y → C

s ♣r❡❞♣✐s♦♠ f1(xi) = ai✱ i = 1, . . . , n✳ ●❧❡❞❡ ♥❛ ♣r❡❞♣♦st❛✈❦♦ ♥✐ t❡➸❦♦

♣r❡✈❡r✐t✐ ❞♦❜r❡ ❞❡✜♥✐r❛♥♦st✐ ♣r❡s❧✐❦❛✈❡ f1✳ P♦t❡♠ ♣♦ ❍❛❤♥ ❇❛♥❛❝❤♦✈❡♠

✐③r❡❦✉ ♦❜st❛❥❛ t❛❦ f ∈ X∗✱ ❞❛ ❥❡ ③♦➸✐t❡✈ f ♥❛ Y r❛✈♥♦ ♣r❡s❧✐❦❛✈❛ f1 t❡r

f = f1 ≤ 1.

✸✹✳ ◆❛❥ ❜♦ X ♥♦r♠✐r❛♥ ♣r♦st♦r✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ ❞✐♠X < ∞ ♥❛t❛♥❦♦ t❡❞❛❥✱

❦♦ ❥❡ ❞✐♠X∗ < ∞✳

◆❛♠✐❣✳ ✭=⇒✮ ❷❡ ❥❡ ❞✐♠X < ∞✱ ❥❡ ❞✐♠X = ❞✐♠X∗✳

✭⇐=✮ Pr❡❞♣♦st❛✈✐♠♦✱ ❞❛ ❥❡ ❞✐♠X = ∞✳ P♦t❡♠ ♦❜st❛❥❛ ♥❡s❦♦♥↔♥❛ ❧✐♥✲

❡❛r♥♦ ♥❡♦❞✈✐s♥❛ ♠♥♦➸✐❝❛ {x1, x2, . . .} ⊂ X✳ ❩❛ ✈s❛❦♦ ♥❛r❛✈♥♦ ➨t❡✈✐❧♦

n ❞❡✜♥✐r❛❥♠♦ ❧✐♥❡❛r❡♥ ❢✉♥❦❝✐♦♥❛❧ fn ∈ X∗ s ♣r❡❞♣✐s♦♠ fn(xn) = 1 ✐♥

fn(xi) = 0✱ ↔❡ ❥❡ i = n✳ ▲✐♥❡❛r♥✐ ❢✉♥❦❝✐♦♥❛❧✐ fn, n ∈ N, s♦ ❧✐♥❡❛r♥♦

♥❡♦❞✈✐s♥✐✳ ❚♦r❡❥ ❥❡ ❞✐♠X∗ = ∞✳

✸✺✳ ◆❛❥ ❜♦ (X, . ) ♥♦r♠✐r❛♥ ♣r♦st♦r ✐♥ x ∈ X t❛❦✱ ❞❛ ❥❡ x = 1✳ P♦❦❛➸✐t❡✱

❞❛ ♦❜st❛❥❛ t❛❦❛ ③❛♣rt❛ ❤✐♣❡rr❛✈♥✐♥❛ H✱ ❞❛ ❥❡ d(x, H) = 1.

◆❛♠✐❣✳ P♦ ♣♦s❧❡❞✐❝✐ ❍❛❤♥ ❇❛♥❛❝❤♦✈❡❣❛ ✐③r❡❦❛ ♦❜st❛❥❛ t❛❦ f ∈ X∗✱ ❞❛

❥❡ f = 1 ✐♥ f(x) = x = 1✳ P♦❦❛➸✐♠♦✱ ❞❛ ③❛ ③❛♣rt♦ ❤✐♣❡rr❛✈♥✐♥♦

Kerf = H ✈❡❧❥❛ d(x, H) = 1. ❑❡r ❥❡ 1 = x = x − 0 ✱ ❥❡ d(x, H) ≤ 1✳

❱ ♥❛❞❛❧❥❡✈❛♥❥✉ ➸❡❧✐♠♦ ♣♦❦❛③❛t✐✱ ❞❛ ❥❡ 1 ≤ d(x, H) = infh∈H x − h ✳

Pr❡❞♣♦st❛✈✐♠♦ ♥❛s♣r♦t♥♦✳ ❙❧❡❞✐✱ ❞❛ ❥❡ f < 1✱ ❦❛r ❥❡ ✈ ♥❛s♣r♦t❥✉ s

♣r❡❞♣♦st❛✈❦♦✳

✸✻✳ ◆❛❥ ❜♦ X r❡✢❡❦s✐✈❡♥ ♣r♦st♦r ✐♥ f ∈ X∗✳ P♦❦❛➸✐t❡✱ ❞❛ ♦❜st❛❥❛ t❛❦ x ∈ X✱

❞❛ ❥❡ x = 1 ✐♥ f(x) = f ✳

◆❛♠✐❣✳ P♦ ♣♦s❧❡❞✐❝✐ ❍❛❤♥ ❇❛♥❛❝❤♦✈❡❣❛ ✐③r❡❦❛ ♦❜st❛❥❛ t❛❦ F ∈ X∗∗✱

❞❛ ❥❡ F (f) = f ✐♥ F = 1✳ ❑❡r ❥❡ X r❡✢❡❦s✐✈❡♥ ♣r♦st♦r✱ ♦❜st❛❥❛ t❛❦

x ∈ X✱ ❞❛ ❥❡ Φ(x) = F ✱ ❦❥❡r ❥❡ Φ : X → X∗∗ s✉r❥❡❦t✐✈♥❛ ❦❛♥♦♥✐↔♥❛

♣r❡s❧✐❦❛✈❛✳ ❚♦r❡❥ ❥❡ f = F (f) = f(x) ✐♥ 1 = F = x ✳

✷✽

✸✼✳ ◆❛❥ ❜♦ X r❡✢❡❦s✐✈❡♥ ♣r♦st♦r ✐♥ Y ③❛♣rt ♣♦❞♣r♦st♦r ♣r♦st♦r❛ X∗✳ P♦❦❛➸✐t❡✱

❞❛ ♦❜st❛❥❛ t❛❦ ♥❡♥✐↔❡❧♥✐ ✈❡❦t♦r x ∈ X✱ ❞❛ ❥❡ f(x) = 0 ③❛ ✈s❛❦ f ∈ Y ✳

◆❛♠✐❣✳ ●❧❡❞❡ ♥❛ ♣♦s❧❡❞✐❝♦ ❍❛❤♥ ❇❛♥❛❝❤♦✈❡❣❛ ✐③r❡❦❛ ♦❜st❛❥❛ t❛❦ F ∈

X∗∗✱ ❞❛ ❥❡ F (Y ) = 0✳ ❑❡r ❥❡ X r❡✢❡❦s✐✈❡♥ ♣r♦st♦r✱ ♦❜st❛❥❛ t❛❦ x ∈ X✱

❞❛ ❥❡ Φ(x) = F ✱ ❦❥❡r ❥❡ Φ : X → X∗∗ s✉r❥❡❦t✐✈♥❛ ❦❛♥♦♥✐↔♥❛ ♣r❡s❧✐❦❛✈❛✳

❚♦r❡❥ ❥❡ F (f) = f(x) = 0 ③❛ ✈s❛❦ f ∈ Y ✳

✸✽✳ ◆❛❥ ❜♦ X ♥♦r♠✐r❛♥ ♣r♦st♦r✱ x ∈ X ✐♥ ♥❛❥ ❜♦ E ⊂ X✳ P♦❦❛➸✐t❡✱ ❞❛ st❛

♥❛s❧❡❞♥❥✐ ❞✈❡ tr❞✐t✈✐ ❡❦✈✐✈❛❧❡♥t♥✐✿

✭❛✮ x ∈ L(E)✱

✭❜✮ ↔❡ ❥❡ f ∈ X∗ t❛❦✱ ❞❛ ❥❡ f(E) = 0✱ s❧❡❞✐ f(x) = 0.

◆❛♠✐❣✳ ✭✭❛✮ ⇒ ✭❜✮✮ ❷❡ ❥❡ x ∈ L(E)✱ ❥❡ x = limn→∞ xn✱ xn ∈ L(E)✳ ◆✐

t❡➸❦♦ ♣r❡✈❡r✐t✐✱ ❞❛ ♣♦t❡♠ s❧❡❞✐ ➸❡❧❡♥♦✳

✭✭❜✮ ⇒ ✭❛✮✮ Pr❡❞♣♦st❛✈✐♠♦✱ ❞❛ x /∈ L(E)✳ P♦t❡♠ ♣♦ ♣♦s❧❡❞✐❝✐ ❍❛❤♥

❇❛♥❛❝❤♦✈❡❣❛ ✐③r❡❦❛ ♦❜st❛❥❛ t❛❦ f ∈ X∗✱ ❞❛ ❥❡ f(L(E)) = 0 ✐♥ f(x) = 0✳

❚♦r❡❥ ❥❡ t✉❞✐ f(E) = 0✳ ●❧❡❞❡ ♥❛ ✭❜✮ s❧❡❞✐✱ ❞❛ ❥❡ f(x) = 0, ❦❛r ❥❡ ✈

♣r♦t✐s❧♦✈❥✉ s ♣r❡❞♣♦st❛✈❦♦✳

✸✾✳ ◆❛❥ ❜♦ X r❡✢❡❦s✐✈❡♥ ♣r♦st♦r ✐♥ ♥❛❥ ❜♦ f ∈ X∗✳

✭❛✮ P♦❦❛➸✐t❡✱ ❞❛ ♦❜st❛❥❛ t❛❦ y ∈ X✱ ❞❛ ❥❡ x − f(x)y ∈ Kerf ③❛ ✈s❛❦

f

x ∈ X✳

✭❜✮ ◆❛❥ ❜♦ ♣r❡s❧✐❦❛✈❛ P : X → X ❞❡✜♥✐r❛♥❛ s ♣r❡❞♣✐s♦♠

f (x)

P x = x −

y, x ∈ X.

f

P♦❦❛➸✐t❡✱ ❞❛ ❥❡ P ♣r♦❥❡❦t♦r✱ ImP = Kerf ✐♥ I − P = 1✳

❘❡➨✐t❡✈✳

✭❛✮ ●❧❡❞❡ ♥❛ ♥❛❧♦❣♦ ✸✻ ♦❜st❛❥❛ t❛❦ y ∈ X✱ ❞❛ ❥❡ f(y) = f ✐♥ 1 = y ✳

P♦t❡♠ ❥❡ f(x − f(x)y) = 0 ③❛ ✈s❛❦ x ∈ X✳

f

✷✾

✭❜✮ ❑❡r ❥❡ P y = y − f(y)y = 0✱ ♥✐ t❡➸❦♦ ♣r❡✈❡r✐t✐✱ ❞❛ ❥❡ P 2 = P ✳

f

◆❛❥ ❜♦ x ∈ ImP ✳ P♦t❡♠ ❥❡ x = P x ∈ Kerf✳ P♦❦❛➸✐♠♦ ➨❡ ♦❜r❛t♥♦✿

↔❡ ❥❡ x ∈ Kerf✱ ❥❡ f(x) = 0 ✐♥ ③❛t♦ ❥❡ x = P x ∈ ImP ✳

❑❡r ❥❡

|f(x)|

x − P x =

y ≤ x ,

f

❥❡ I − P ≤ 1✳ ❯♣♦➨t❡✈❛❥♠♦✱ ❞❛ ❥❡ y − P y = 1✱ ✐③ ↔❡s❛r s❧❡❞✐

I − P = 1✳

✹✵✳ ◆❛❥ ❜♦ X ❇❛♥❛❝❤♦✈ ♣r♦st♦r✳ ❩❛ ♣♦❧❥✉❜♥❛ y, z ∈ X ✐♥ f, g ∈ X∗ ❞❡✜♥✐✲

r❛❥♠♦ P : X → X s ♣r❡❞♣✐s♦♠ P x = f(x)y + g(z)x✳

✭❛✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ P ∈ B(X)✳

✭❜✮ ❉❡♥✐♠♦✱ ❞❛ ❥❡ X r❡✢❡❦s✐✈❡♥ ♣r♦st♦r✱ Y ∗ ♣r❛✈✐ ③❛♣rt ♣♦❞♣r♦st♦r

♣r♦st♦r❛ X∗ ✐♥ ❞❛ ❥❡ f = 1✳ P♦❦❛➸✐t❡✱ ❞❛ ♦❜st❛❥❛t❛ t❛❦❛ ♥❡♥✐↔❡❧♥❛

y, z ∈ X✱ ❞❛ ❥❡ P ♣r♦❥❡❦t♦r ③❛ ❦❛t❡r✐❦♦❧✐ g ∈ Y ∗✳

❘❡➨✐t❡✈✳ ✭❜✮ ❱✐❞✐♠♦✱ ❞❛ ❥❡

P 2x = P (f (x)y + g(z)x) = f (f (x)y + g(z)x)y + g(z)(f (x)y + g(z)x)

= f (x)f (y)y + g(z)f (x)y + g(z)f (x)y + g(z)g(z)x.

●❧❡❞❡ ♥❛ ♥❛❧♦❣♦ ✸✻ ♦❜st❛❥❛ t❛❦ y ∈ X✱ ❞❛ ❥❡ y = 1 ✐♥ f(y) = f = 1✳

❷❡ ✉♣♦➨t❡✈❛♠♦ ➨❡ ♥❛❧♦❣♦ ✸✼✱ ♦❜st❛❥❛ t❛❦ z ∈ X✱ ❞❛ ❥❡ g(z) = 0 ③❛ ✈s❛❦

g ∈ Y ∗✳ ■③ t❡❣❛ s❧❡❞✐ ➸❡❧❡♥♦✳

✹✶✳ ◆❛❥ ❜♦ X ❇❛♥❛❝❤♦✈ ♣r♦st♦r ✐♥ ♥❛❥ ❜♦st❛ A, B ∈ B(X) t❛❦❛ ♦♣❡r❛t♦r❥❛✱

❞❛ ❥❡ AT B = BT A ③❛ ✈s❛❦ T ∈ B(X)✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ B = λA ③❛ ♥❡❦

λ ∈ C✳

❘❡➨✐t❡✈✳ ◆❛❥ ❜♦ 0 = f ∈ X∗ ✐♥ ♥❛❥ ❜♦ T : X → X ♦♣❡r❛t♦r ❞❡✜♥✐r❛♥

s ♣r❡❞♣✐s♦♠ T (x) = f(x)u ③❛ ♥❡❦ 0 = u ∈ X✳ P♦t❡♠ ❥❡ T ∈ B(X)✳ P♦

❡♥✐ str❛♥✐ ❥❡ AT Bx = f(Bx)Au✱ ♣♦ ❞r✉❣✐ str❛♥✐ ♣❛ BT Ax = f(Ax)Bu✳

❑❡r ❥❡ AT B = BT A, s❧❡❞✐ ➸❡❧❡♥♦✳

✸✵

P♦❣❧❛✈❥❡ ✸

❍✐❧❜❡rt♦✈✐ ♣r♦st♦r✐

✶✳ ◆❛❥ ❜♦ X ❍✐❧❜❡rt♦✈ ♣r♦st♦r✱ Y ③❛♣rt ♣♦❞♣r♦st♦r ✐♥ f ∈ X∗✳ P♦❦❛➸✐t❡✱

❞❛ ♦❜st❛❥❛t❛ t❛❦❛ ❧✐♥❡❛r♥❛ ❢✉♥❦❝✐♦♥❛❧❛ f1, f2 ∈ X∗✱ ❞❛ ✈❡❧❥❛✿

✭❛✮ f = f1 + f2,

✭❜✮ f1(Y ⊥) = {0} ✐♥ f2(Y ) = {0},

✭❝✮ f 2 = f 2

2

1

+ f2 .

◆❛♠✐❣✳ ❱❡♠♦✱ ❞❛ ❥❡ X = Y ⊕ Y ⊥✳ P♦ ❘✐❡s③♦✈❡♠ ✐③r❡❦✉ ♦❜st❛❥❛ t❛❦

a = a1 + a2 ∈ X✱ a1 ∈ Y ✱ a2 ∈ Y ⊥✱ ❞❛ ❥❡ f(x) = x, a ③❛ ✈s❛❦ x ∈ X ✐♥

f = a ✳ ❋✉♥❦❝✐❥✐ f1 ✐♥ f2 ❞❡✜♥✐r❛♠♦ s ♣r❡❞♣✐s♦♠❛ f1(x) = x, a1 t❡r

f2(x) = x, a2 ✳

✷✳ ◆❛❥ ❜♦ S ♣♦❞♣r♦st♦r ❍✐❧❜❡rt♦✈❡❣❛ ♣r♦st♦r❛ X✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡

(S⊥)⊥ = L(S).

◆❛♠✐❣✳ ◆❛❥ ❜♦st❛ x ∈ S⊥ ✐♥ y ∈ L(S)✳ ❚♦r❡❥ ❥❡ y = ❧✐♠n→∞sn✱

⊥

sn ∈ L(S)✳ P♦t❡♠ ❥❡ x, y = 0✳ ■③ t❡❣❛ s❧❡❞✐✱ ❞❛ ❥❡ S⊥ ⊆ L(S) . ❑❡r ❥❡

⊥

⊥

S ⊆ L(S)✱ ❥❡ L(S) ⊆ S⊥✱ ✐③ ↔❡s❛r s❧❡❞✐ (S⊥)⊥ = (L(S) )⊥ = L(S)✳

✸✳ ◆❛❥ ❜♦ H ❍✐❧❜❡rt♦✈ ♣r♦st♦r ✐♥ ♥❛❥ ❜♦ A ∈ B(H)✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡

H = KerA ⊕ ImA∗.

✸✶

◆❛♠✐❣✳ ❑❡r ❥❡ ImA∗ ⊆ ImA∗✱ s❧❡❞✐ (ImA∗)⊥ ⊆ (ImA∗)⊥✳

◆❛❥ ❜♦ x ∈ KerA ✐♥ ♥❛❥ ❜♦ y ∈ ImA∗✳ P♦t❡♠ ❥❡ y = limn→∞ A∗yn✱

{A∗yn}n∈N ⊂ ImA∗✳ ❑❡r ❥❡ x, A∗yn = Ax, yn = 0✱ s❧❡❞✐ x ∈ (ImA∗)⊥✳

❚♦r❡❥ ❥❡ KerA ⊆ (ImA∗)⊥✳

◆❛❥ ❜♦ x ∈ (ImA∗)⊥✳ P♦t❡♠ ❥❡ 0 = x, A∗y = Ax, y ③❛ ✈s❛❦ y ∈ H✳

■③ t❡❣❛ s❧❡❞✐✱ ❞❛ ❥❡ Ax = 0✱ ❦❛r ♣♦♠❡♥✐ (ImA∗)⊥ ⊆ KerA✳ ❙ t❡♠ s♠♦

♣♦❦❛③❛❧✐✱ ❞❛ ❥❡ (ImA∗)⊥ = KerA✳ ❑❡r ❥❡ H = (ImA∗)⊥ ⊕ ImA∗✱ s❧❡❞✐

➸❡❧❡♥♦✳

✹✳ ◆❛❥ ❜♦st❛ X ✐♥ Y ③❛♣rt❛ ♣♦❞♣r♦st♦r❛ ❍✐❧❜❡rt♦✈❡❣❛ ♣r♦st♦r❛ H✳ P♦❦❛➸✐t❡

❡♥❛❦♦st

(X⊥ + Y ⊥)⊥ = X ∩ Y.

❆❧✐ ✈❡❧❥❛ tr❞✐t❡✈ t✉❞✐ ✈ ♣r✐♠❡r✉✱ ❦♦ ♣r♦st♦r❛ ♥✐st❛ ③❛♣rt❛❄

◆❛♠✐❣✳ ◆❛❥ ❜♦ x ∈ (X⊥ + Y ⊥)⊥✳ P♦t❡♠ ❥❡ x, x1 + x2 = 0✱ x1 ∈ X⊥✱

x2 ∈ Y ⊥✳ ■③ t❡❣❛ s❧❡❞✐✱ ❞❛ ❥❡ x ∈ X ∩ Y ✳

❷❡ ❥❡ x ∈ X ∩ Y ✱ ❥❡ x, x1 + x2 = 0 ③❛ ✈s❛❦ x1 ∈ X⊥ ✐♥ x2 ∈ Y ⊥✳ ❚♦r❡❥

❥❡ x ∈ (X⊥ + Y ⊥)⊥✳

P♦❦❛③❛♥❛ ❡♥❛❦♦st ♥❡ ✈❡❧❥❛ ✈ ♣r✐♠❡r✉✱ ❦♦ ♣r♦st♦r❛ X ✐♥ Y ♥✐st❛ ③❛♣rt❛✳

Pr✐♠❡r✿ X = Y ❥❡ ♣♦❞♣r♦st♦r t❛❦✐❤ ③❛♣♦r❡❞✐❥ ✐③ l2✱ ❦✐ ✐♠❛❥♦ ❦✈❡↔❥❡♠✉

❦♦♥↔♥♦ ♠♥♦❣♦ ♥❡♥✐↔❡❧♥✐❤ ↔❧❡♥♦✈✳

✺✳ ◆❛❥ ❜♦ H ❍✐❧❜❡rt♦✈ ♣r♦st♦r ✐♥ ♥❛❥ ❜♦ P ∈ B(H) ♣r♦❥❡❦t♦r✳ P♦❦❛➸✐t❡✱

❞❛ ❥❡ ImP ③❛♣rt ♣♦❞♣r♦st♦r✳ P♦❦❛➸✐t❡ t✉❞✐ ✐♠♣❧✐❦❛❝✐❥✐ (a) =⇒ (b) t❡r

(b) =⇒ (c)✿

✭❛✮ P = P ∗✱

✭❜✮ ImP = (KerP )⊥✱

✭❝✮ P x, x = P x 2 ③❛ ✈s❛❦ x ∈ H✳

◆❛♠✐❣✳ ❑❡r ❥❡ ImP = Ker(I − P )✱ ❥❡ ImP ③❛♣rt ♣r♦st♦r✳

✭✭❛✮ ⇒ ✭❜✮✮ ◆❛❥♣r❡❥ ♣♦❦❛➸❡♠♦✱ ❞❛ ❥❡ KerP = (ImP )⊥✱ ✐③ ↔❡s❛r s❧❡❞✐

➸❡❧❡♥♦✿ (KerP )⊥ = (ImP )⊥⊥ = ImP.

✸✷

✭✭❜✮ ⇒ ✭❝✮✮ ❑❡r ❥❡ H = ImP ⊕ (ImP )⊥✱ ❥❡ H = ImP ⊕ KerP ✳ ◆❛❥ ❜♦

x ∈ H✳ P♦t❡♠ ❧❛❤❦♦ x ③❛♣✐➨❡♠♦ ❦♦t x = y + z✱ ❦❥❡r st❛ y ∈ KerP ✐♥

z ∈ ImP ✳ ❚♦r❡❥ ❥❡ P x 2 = P x, P x = P z, P z = z, P z = x, P x .

✻✳ ◆❛❥ ❜♦ H ❍✐❧❜❡rt♦✈ ♣r♦st♦r ✐♥ A ∈ B(H) t❛❦ s❡❜✐❛❞❥✉♥❣✐r❛♥ ♦♣❡r❛t♦r✱

❞❛ ❥❡ x ≤ Ax ③❛ ✈s❛❦ x ∈ H✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ (ImA)⊥ = 0 ✐♥ ♦❞ t♦❞

✐③♣❡❧❥✐t❡✱ ❞❛ ❥❡ A ❤♦♠❡♦♠♦r✜③❡♠✳

❘❡➨✐t❡✈✳ ◆❛❥ ❜♦ x ∈ (ImA)⊥✳ P♦t❡♠ ❥❡ 0 = x, Ay = Ax, y ③❛ ✈s❛❦

y ∈ H✳ ■③ t❡❣❛ s❧❡❞✐✱ ❞❛ ❥❡ x ≤ Ax = 0✳ ❚♦r❡❥ ❥❡ x = 0✳

❱ ♥❛❞❛❧❥❡✈❛♥❥✉ ♣♦❦❛➸✐♠♦✱ ❞❛ ❥❡ ImA ③❛♣rt ♣r♦st♦r✳ ◆❛❥ ❜♦ {Axn}n∈N

❦♦♥✈❡r❣❡♥t♥♦ ③❛♣♦r❡❞❥❡ ✈ ♣r♦st♦r✉ ImA ③ ❧✐♠✐t♦ x✳ ●❧❡❞❡ ♥❛ ♣r❡❞✲

♣♦st❛✈❦♦✱ ❞❛ ❥❡ x ≤ Ax ③❛ ✈s❛❦ x ∈ H✱ ❥❡ {xn}n∈N ❈❛✉❝❤②❥❡✈♦ ③❛✲

♣♦r❡❞❥❡ ✈ H ✐♥ ③❛t♦ ❦♦♥✈❡r❣❡♥t♥♦ ③ ❧✐♠✐t♦ y✳ ❚♦r❡❥ ❥❡ x = limn→∞ Axn =

Ay ∈ ImA✳

●❧❡❞❡ ♥❛ ♣♦❦❛③❛♥♦ ❥❡ H = ImA ⊕ (ImA)⊥ = ImA✳ Pr❛✈ t❛❦♦ ♥✐ t❡➸❦♦

♣r❡✈❡r✐t✐ ✐♥❥❡❦t✐✈♥♦st✐ ♦♣❡r❛t♦r❥❛ A✳ P♦ ✐③r❡❦✉ ♦ ♦❞♣rt✐ ♣r❡s❧✐❦❛✈✐ s❧❡❞✐✱

❞❛ ❥❡ A−1 ③✈❡③❡♥ ♦♣❡r❛t♦r✳ ❚♦r❡❥ ❥❡ A ❤♦♠❡♦♠♦r✜③❡♠✳

✼✳ ◆❛❥ ❜♦ H ❍✐❧❜❡rt♦✈ ♣r♦st♦r✱ A, B ∈ B(H) ✐♥ ♥❛❥ ❜♦ C = A∗A + B∗B✳

P♦❦❛➸✐t❡✿

✭❛✮ KerC = KerA ∩ KerB✳

✭❜✮ ▲❛st♥❡ ✈r❡❞♥♦st✐ ♦♣❡r❛t♦r❥❛ C s♦ ♥❡♥❡❣❛t✐✈♥❛ r❡❛❧♥❛ ➨t❡✈✐❧❛✳

◆❛♠✐❣✳

✭❛✮ ◆❛❥ ❜♦ x ∈ KerC✳ P♦t❡♠ ❥❡ 0 = A∗Ax, y + B∗Bx, y ③❛ ✈s❛❦

y ∈ H✳ ❙❧❡❞✐ Ax 2 + Bx 2 = 0 ✐♥ ③❛t♦ ❥❡ Ax = Bx = 0✳ ❚♦r❡❥ ❥❡

x ∈ KerA ∩ KerB✳

❷❡ ❥❡ x ∈ KerA∩KerB✱ ❥❡ (A∗A+B∗B)x = 0 ✐♥ ③❛t♦ ❥❡ x ∈ KerC.

✭❜✮ ◆❛❥ ❜♦ λ ❧❛st♥❛ ✈r❡❞♥♦st ♦♣❡r❛t♦r❥❛ C✳ ❚♦r❡❥ ♦❜st❛❥❛ t❛❦ 0 = x ∈

H✱ ❞❛ ❥❡ Cx = λx✳ ■③ t❡❣❛ s❧❡❞✐✱ ❞❛ ❥❡ λ x, x = λ x, x ✳ ❑❡r ❥❡

Cx, x ≥ 0✱ ❥❡ λ ≥ 0✳

✸✸

✽✳ ◆❛❥ ❜♦ U ∈ B(H) ✉♥✐t❛r♥✐ ♦♣❡r❛t♦r ✐♥ P ∈ B(H) ♦rt♦❣♦♥❛❧♥✐ ♣r♦❥❡❦t♦r

♥❛ ❍✐❧❜❡rt♦✈❡♠ ♣r♦st♦r✉ (H, ., . )✳ ❱♣❡❧❥✐♠♦ {., .} : H × H → H s

♣r❡❞♣✐s♦♠ {x, y} = Ux+P Ux, Uy ✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ (H, {., .}) ❍✐❧❜❡rt♦✈

♣r♦st♦r✳

◆❛♠✐❣✳ ◆✐ t❡➸❦♦ ♣r❡✈❡r✐t✐ ❞❛ ❥❡ {., .} s❦❛❧❛r♥✐ ♣r♦❞✉❦t ♥❛ ♣r♦st♦r✉ H✳

P♦t❡♠ ❥❡ s ♣r❡❞♣✐s♦♠ {x, x} = x 2✱1 x ∈ H✱ ❞❡✜♥✐r❛♥❛ ♥♦r♠❛ ♥❛ H✳

❑❡r st❛ ♥♦r♠✐ . ✐♥ . 1 ❡❦✈✐✈❛❧❡♥t♥✐✱

x 2 = x, x ≤ Ux 2 + P Ux 2 = x 21 ≤ ( U 2 + P U 2) x ,

❥❡ (H, {., .}) ❍✐❧❜❡rt♦✈ ♣r♦st♦r✳

✾✳ ◆❛❥ ❜♦ ♦♣❡r❛t♦r P : l2 → l2 ♣♦❞❛♥ s ♣r❡❞♣✐s♦♠

P (x1, x2, x3, x4 . . .) = (0, x2, 0, x4, 0, . . . , 0, x2n, 0, . . .).

P♦❦❛➸✐t❡✱ ❞❛ ❥❡ P ♦rt♦❣♦♥❛❧♥✐ ♣r♦❥❡❦t♦r✳ P♦✐➨↔✐t❡ KerP ✱ (KerP )⊥ ✐♥

ImP ✳

❘❡➨✐t❡✈✳ ◆✐ t❡➸❦♦ ♣r❡✈❡r✐t✐✱ ❞❛ ❥❡ P ∈ B(l2) ✐♥ P 2 = P = P ∗✱ ♣r✐

↔❡♠❡r ✉♣♦➨t❡✈❛♠♦✱ ❞❛ ❥❡ P x, y = x, P ∗y ✱ x, y ∈ l2✳ ❱✐❞✐♠♦✱ ❞❛ ❥❡

ImP = KerP ✐♥

KerP = {(x1, x2, x3, x4, . . .) | x2n = 0, ∀n ∈ N},

(KerP )⊥ = {(x1, x2, x3, x4, . . .) | x2n−1 = 0, ∀n ∈ N}.

✶✵✳ ❩❛ ✈s❛❦♦ ♥❛r❛✈♥♦ ➨t❡✈✐❧♦ n ❞❡✜♥✐r❛♠♦ ❢✉♥❦❝✐♦♥❛❧ fn ♥❛ ♣r♦st♦r✉ l2 s

♣r❡❞♣✐s♦♠

x

x

f

2

n

n(x1, x2, x3, . . .) = x1 + √

+ . . . + √ .

2

n

P♦❦❛➸✐t❡✱ ❞❛ ❥❡ fn ♦♠❡❥❡♥ ❧✐♥❡❛r❡♥ ❢✉♥❦❝✐♦♥❛❧ ✐♥ ✐③r❛↔✉♥❛❥t❡ ♥❥❡❣♦✈♦

♥♦r♠♦✳ ❆❧✐ ❥❡ ③❛♣♦r❡❞❥❡ {fn}n∈N ❦♦♥✈❡r❣❡♥t♥♦❄

❘❡➨✐t❡✈✳ ❑❡r ❥❡

1

1

fn(x1, x2, x3, . . .) = (x1, x2, x3, . . .), (1, √ , . . . , √ , 0, 0, . . .) ,

2

n

✸✹

❥❡ fn = (1, 1√ , . . . , 1

√ , 0, 0, . . .) =

1 + 1 + . . . + 1 ✭❘✐❡s③♦✈ ✐③r❡❦✮✳

2

n

2

n

◆❛❥ ❜♦ an = (1, 1√ , . . . , 1

√ , 0, 0, . . .). ❱✐❞✐♠♦✱ ❞❛ ❥❡

2

n

1

1

1

f

2

2

n − fm

= an − am

=

+

+ . . . +

.

n

n − 1

m + 1

■③ t❡❣❛ s❧❡❞✐✱ ❞❛ ③❛♣♦r❡❞❥❡ {fn}n∈N ♥✐ ❦♦♥✈❡r❣❡♥t♥♦✳

✶✶✳ ❩❛ ✈s❛❦♦ ♥❛r❛✈♥♦ ➨t❡✈✐❧♦ n ❞❡✜♥✐r❛♠♦ ❢✉♥❦❝✐♦♥❛❧ fn ♥❛ ♣r♦st♦r✉ l2 s

♣r❡❞♣✐s♦♠

1

1

1

1

fn(x1, x2, x3, . . .) =

x

x

x

x

1

1 +

2

2 + . . . +

n

n−1 +

n

n.

3

−1

2

32

3 2





3 2


P♦❦❛➸✐t❡✱ ❞❛ ❥❡ fn ♦♠❡❥❡♥ ❧✐♥❡❛r❡♥ ❢✉♥❦❝✐♦♥❛❧ ✐♥ ✐③r❛↔✉♥❛❥t❡ ♥❥❡❣♦✈♦

♥♦r♠♦✳ ❆❧✐ ❥❡ ③❛♣♦r❡❞❥❡ {fn}n∈N ❦♦♥✈❡r❣❡♥t♥♦❄

◆❛♠✐❣✳ ❩❛♣♦r❡❞❥❡ {fn}n∈N ❥❡ ❦♦♥✈❡r❣❡♥t♥♦✳

✶✷✳ ◆❛❥ ❜♦ U ∈ B(H) ✉♥✐t❛r❡♥ ✐♥ A ∈ B(H) ♣♦❧❥✉❜❡♥ ♦♣❡r❛t♦r ♥❛ ❍✐❧❜❡r✲

t♦✈❡♠ ♣r♦st♦r✉ H✳ P♦❦❛➸✐t❡✿

✭❛✮ UAx = Ax ③❛ ✈s❛❦ x ∈ H ✐♥ UA = A ✱

✭❜✮ Ux = x ③❛ ✈s❛❦ x ∈ H ✐♥ AU = A ✳

◆❛♠✐❣✳

✭❛✮ ❱✐❞✐♠♦✱ ❞❛ ❥❡ UAx 2 = UAx, UAx = Ax, Ax = Ax 2✱ ✐③

↔❡s❛r s❧❡❞✐ UA = A ✳

✭❜✮ ❑❡r ❥❡ AUx ≤ A Ux = A , ❥❡ AU ≤ A ✳ P♦ ❞r✉❣✐ str❛♥✐ ♣❛

x

x

♦♣❛③✐♠♦✱ ❞❛ ❥❡ Ax = AU(U∗x) ≤ AU ✳ ❚♦r❡❥ ❥❡ A ≤ AU ✳

x

U ∗x

✶✸✳ ◆❛❥ ❜♦ H r❡❛❧❡♥ ❍✐❧❜❡rt♦✈ ♣r♦st♦r ✐♥ ♥❛❥ ❜♦ T : H → H ❧✐♥❡❛r❡♥ ♦♣✲

❡r❛t♦r✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ T x, x = 0 ③❛ ✈s❛❦ x ∈ H ♥❛t❛♥❦♦ t❡❞❛❥✱ ❦♦ ❥❡

T = −T ∗✳

✸✺

❘❡➨✐t❡✈✳ ✭=⇒) ❱✐❞✐♠♦✱ ❞❛ ❥❡ T (x + y), x + y = 0 ③❛ ✈s❛❦ x, y ∈ H✳

❙❧❡❞✐ 0 = T x, y + T y, x = x, T ∗y + x, T y = x, (T ∗ + T )y , ❦❛r ♥❛s

♣r✐✈❡❞❡ ❞♦ ➸❡❧❡♥❡ ❡♥❛❦♦st✐✳

✭⇐=✮ ■③ ❡♥❛❦♦st✐ T x, x = x, T ∗x = − T x, x s❧❡❞✐ ➸❡❧❡♥♦✳

✶✹✳ ◆❛❥ ❜♦ H ❦♦♠♣❧❡❦s❡♥ ❍✐❧❜❡rt♦✈ ♣r♦st♦r ✐♥ ♥❛❥ ❜♦ T : H → H ❧✐♥❡❛r❡♥

♦♣❡r❛t♦r✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ T x, x = 0 ③❛ ✈s❛❦ x ∈ H ♥❛t❛♥❦♦ t❡❞❛❥✱ ❦♦

❥❡ T = 0✳

❘❡➨✐t❡✈✳ ✭=⇒✮ ❙ ♣♦♠♦↔❥♦ ❧✐♥❡❛r✐③❛❝✐❥❡ ✭♥❛♠❡st♦ x ♣✐➨✐♠♦ x+y✱ y ∈ H✮

✈✐❞✐♠♦✱ ❞❛ ❥❡ 0 = T (x + y), x + y = T x, y + T y, x . ❷❡ ♥❛♠❡st♦ x

♣✐➨❡♠♦ ix✱ ❥❡ 0 = T ix, y + T y, ix . ■③ t❡❣❛ s❧❡❞✐ 0 = i T x, y −i T y, x .

❚♦r❡❥ ❥❡ 0 = T x, y − T y, x . ❷❡ s❡➨t❡❥❡♠♦ ♣r✈♦ ✐♥ ③❛❞♥❥♦ ❡♥❛❦♦st✱

❞♦❜✐♠♦ T x, y = 0 ③❛ ✈s❡ x, y ∈ H✳ ■③ t❡❣❛ s❧❡❞✐ ➸❡❧❡♥♦✱ T = 0✳

✶✺✳ ◆❛❥ ❜♦ H ❦♦♠♣❧❡❦s❡♥ ❍✐❧❜❡rt♦✈ ♣r♦st♦r ✐♥ ♥❛❥ ❜♦ T : H → H t❛❦

❧✐♥❡❛r❡♥ ♦♣❡r❛t♦r✱ ❞❛ ❥❡ T x, x ≥ 0 ③❛ ✈s❛❦ x ∈ H✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡

T ∗ = T ✳

❘❡➨✐t❡✈✳ ❑❡r ❥❡ T x, x ≥ 0✱ ❥❡ T x, x = x, T x = x, T x ③❛ ✈s❛❦

x ∈ H✳ P♦ ❞r✉❣✐ str❛♥✐ ♣❛ ❥❡ T x, x = x, T ∗x ✳ ■③ t❡❣❛ s❧❡❞✐✱ ❞❛ ❥❡

(T − T ∗)x, x = 0 ③❛ ✈s❛❦ x ∈ H✳ ●❧❡❞❡ ♥❛ ♣r❡❥➨♥❥♦ ♥❛❧♦❣♦ ❥❡ T = T ∗✳

✶✻✳ ◆❛❥ ❜♦ H ❦♦♠♣❧❡❦s❡♥ ❍✐❧❜❡rt♦✈ ♣r♦st♦r✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ N ∈ B(H)

♥♦r♠❛❧❡♥ ♦♣❡r❛t♦r ♥❛t❛♥❦♦ t❡❞❛❥✱ ❦♦ ❥❡ Nx = N∗x ③❛ ✈s❛❦ x ∈ H✳

◆❛♠✐❣✳ ✭=⇒✮ ❖↔✐t♥♦ ❥❡ Nx 2 = N∗Nx, x = NN∗x, x = N∗x 2

③❛ ✈s❛❦ x ∈ H✳

✭⇐=✮ ●❧❡❞❡ ♥❛ ♣r❡❞♣♦st❛✈❦♦ ♥✐ t❡➸❦♦ ♣♦❦❛③❛t✐✱ ❞❛ ❥❡

(N ∗N − NN∗)x, x = 0

③❛ ✈s❛❦ x ∈ H✳ ❙ ♣♦♠♦↔❥♦ ♥❛❧♦❣❡ ✶✹ s❧❡❞✐ ➸❡❧❡♥♦✱ N∗N = NN∗✳

✸✻

✶✼✳ ◆❛❥ ❜♦ H ❍✐❧❜❡rt♦✈ ♣r♦st♦r ✐♥ ♥❛❥ ❜♦st❛ x, y ∈ H t❛❦❛✱ ❞❛ ❥❡

x + y = x + y .

P♦❦❛➸✐t❡✱ ❞❛ ❥❡ x y = | x, y |✳

◆❛♠✐❣✳ ❖↔✐t♥♦ ❥❡ x + y 2 = x 2 + y 2 + 2 x y ✳ ■③ t❡❣❛ s❧❡❞✐

x, y + y, x = 2 x y ✳ ❚♦r❡❥ ❥❡ | x, y | ≤ x y = ❘❡ x, y ≤ | x, y |✳

✶✽✳ ◆❛❥ ❜♦ (X, ., . ) r❡❛❧❡♥ ❛❧✐ ❦♦♠♣❧❡❦s❡♥ ✈❡❦t♦rs❦✐ ♣r♦st♦r s s❦❛❧❛r♥✐♠

♣r♦❞✉❦t♦♠✳ ❉❡♥✐♠♦✱ ❞❛ ③❛ ✈s❛❦ f ∈ X∗ ♦❜st❛❥❛ t❛❦ y ∈ X✱ ❞❛ ❥❡

f (x) = x, y ③❛ ✈s❛❦ x ∈ X✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ X ♣♦❧♥ ♣r♦st♦r✳

◆❛♠✐❣✳ ◆❛❥ ❜♦ ϕ : X → X∗ ♣r❡s❧✐❦❛✈❛ ❞❡✜♥✐r❛♥❛ s ♣r❡❞♣✐s♦♠ ϕ(a) =

fa✱ ❦❥❡r ❥❡ fa(x) = x, a ③❛ ✈s❛❦ x ∈ X✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ ϕ ✐③♦♠❡tr✐❥❛ ✐♥

✉♣♦➨t❡✈❛❥t❡✱ ❞❛ ❥❡ X∗ ♣♦❧♥ ♣r♦st♦r✳

✶✾✳ ◆❛❥ ❜♦ A ❇❛♥❛❝❤♦✈❛ ❛❧❣❡❜r❛✳

✭❛✮ ◆❛❥ ❜♦ a ∈ A ♥✐❧♣♦t❡♥t❡♥✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ σ(a) = {0}✳

✭❜✮ ◆❛❥ ❜♦ e ∈ A ✐❞❡♠♣♦t❡♥t ✐♥ e = 0, 1✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ σ(e) = {0, 1}✳

◆❛♠✐❣✳

✭❛✮ ❑❡r ♦❜st❛❥❛ t❛❦ n ∈ N✱ ❞❛ ❥❡ an−1 = 0 ✐♥ an = 0✱ ❥❡ a ❞❡❧✐t❡❧❥ ♥✐↔❛✳

❩❛t♦ a ♥✐ ♦❜r♥❧❥✐✈✳ ❚♦r❡❥ ❥❡ 0 ∈ σ(a)✳ ◆❛❥ ❜♦ 0 = λ ∈ C. P♦❦❛➸✐t❡✱

❞❛ ♦❜st❛❥❛ ✐♥✈❡r③ ❡❧❡♠❡♥t❛ a−λ1 ✭❣❡♦♠❡tr✐❥s❦❛ ✈rst❛✮✱ ✐③ ↔❡s❛r s❧❡❞✐

➸❡❧❡♥♦✳

✭❜✮ ❑❡r ❥❡ e(e − 1) = 0✱ e = 0, 1✱ ❡❧❡♠❡♥t❛ e ✐♥ e − 1 ♥✐st❛ ♦❜r♥❧❥✐✈❛✳

P♦t❡♠ ♣❛ ❥❡ 0, 1 ∈ σ(e)✳ ◆❛❥ ❜♦ 0, 1 = λ ∈ C. ❑❡r ❥❡ e − λ1 ♦❜r♥❧❥✐✈

❡❧❡♠❡♥t ③ ✐♥✈❡r③♦♠ (e − λ1)−1 = 1(−1 + e )✱ s❧❡❞✐ ➸❡❧❡♥♦✳

λ

1−λ

✸✼

✷✵✳ ◆❛❥ ❜♦ A ❇❛♥❛❝❤♦✈❛ ❛❧❣❡❜r❛ ✐♥ a, b ∈ A✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡

σ(ab) − {0} = σ(ba) − {0}.

◆❛♠✐❣✳ ◆❛❥ ❜♦ λ = 0✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ ab − λ ♦❜r♥❧❥✐✈ ❡❧❡♠❡♥t ♥❛t❛♥❦♦

t❡❞❛❥✱ ❦♦ ❥❡ ba − λ ♦❜r♥❧❥✐✈ ❡❧❡♠❡♥t✳ ❑♦♥❦r❡t♥♦✱ ↔❡ ❥❡ ba − 1 ♦❜r♥❧❥✐✈

❡❧❡♠❡♥t✱ ♣♦t❡♠ ❥❡

(ab − 1)−1 = −(1 + a(1 − ba)−1b).

✷✶✳ ◆❛❥ ❜♦ A ❇❛♥❛❝❤♦✈❛ ❛❧❣❡❜r❛ ③ ❡♥♦t♦ e ✐♥ ♥❛❥ ❜♦ x ∈ A ♦❜r♥❧❥✐✈✳ P♦❦❛➸✐t❡✱

❞❛ ❥❡ λ ∈ σ(x) ♥❛t❛♥❦♦ t❡❞❛❥✱ ❦♦ ❥❡ λ−1 ∈ σ(x−1)✳

◆❛♠✐❣✳ ✭=⇒✮ ◆❛❥ ❜♦ λ ∈ σ(x)✳ Pr❡❞♣♦st❛✈✐♠♦✱ ❞❛ ❥❡ λ−1 /∈ σ(x−1)✳

P♦t❡♠ ❥❡ x−1 − λ−1e ♦❜r♥❧❥✐✈ ❡❧❡♠❡♥t✳ ❑❡r ❥❡ λe − x = λx(x−1 − λ−1e)✱

❥❡ λe − x ♦❜r♥❧❥✐✈ ❡❧❡♠❡♥t✳ ❙ t❡♠ s♠♦ ♣♦❦❛③❛❧✐ ➸❡❧❡♥♦✳

✷✷✳ ◆❛❥ ❜♦ A ❇❛♥❛❝❤♦✈❛ ❛❧❣❡❜r❛ ③ ❡♥♦t♦ e✱ e = 1 ✐♥ ♥❛❥ ❜♦ ϕ : A → C

♥❡♥✐↔❡❧♥ ♠✉❧t✐♣❧✐❦❛t✐✈❡♥ ❧✐♥❡❛r❡♥ ❢✉♥❦❝✐♦♥❛❧✳ P♦❦❛➸✐t❡✿

✭❛✮ ϕ(e) = 1,

✭❜✮ ϕ(a) ∈ σ(a) ③❛ ✈s❛❦ a ∈ A,

✭❝✮ ϕ = 1✳

❘❡➨✐t❡✈✳

✭❛✮ ❑❡r ❥❡ ϕ(a) = ϕ(ae) = ϕ(a)ϕ(e) ③❛ ✈s❛❦ 0 = a ∈ A✱ ❥❡ ϕ(e) = 1.

✭❜✮ ❱✐❞✐♠♦✱ ❞❛ ❥❡ ϕ(a − ϕ(a)e) = ϕ(a) − ϕ(a) = 0✳ ■③ t❡❣❛ s❧❡❞✐✱ ❞❛

a − ϕ(a)e ♥✐ ♦❜r♥❧❥✐✈ ❡❧❡♠❡♥t ✐♥ ③❛t♦ ❥❡ ϕ(a) ∈ σ(a)✳

✭❝✮ ❑❡r ❥❡ ϕ(e) = 1✱ ❥❡ 1 ≤ ϕ ✳ ❙ ♣♦♠♦↔❥♦ t♦↔❦❡ ✭❜✮ ✈✐❞✐♠♦✱ ❞❛ ❥❡

|ϕ(a)| ≤ a ③❛ ✈s❛❦ a ∈ A, ✐③ ↔❡s❛r s❧❡❞✐ ϕ ≤ 1✳

✸✽

✷✸✳ ◆❛❥ ❜♦ X ❦♦♠♣❧❡❦s♥❛ ❇❛♥❛❝❤♦✈❛ ❛❧❣❡❜r❛ ③ ❡♥♦t♦ 1 ✐♥ ♥❛❥ ❜♦ ϕ : X → X

t❛❦❛ ❧✐♥❡❛r♥❛ ♣r❡s❧✐❦❛✈❛✱ ❞❛ ❥❡ ϕ(1) = 1✳ P♦❦❛➸✐t❡✱ ❞❛ st❛ ♥❛s❧❡❞♥❥✐ ❞✈❡

tr❞✐t✈✐ ❡❦✈✐✈❛❧❡♥t♥✐✿

✭❛✮ x ∈ X ❥❡ ♦❜r♥❧❥✐✈ ♥❛t❛♥❦♦ t❡❞❛❥✱ ❦♦ ❥❡ ϕ(x) ♦❜r♥❧❥✐✈✱

✭❜✮ σ(ϕ(x)) = σ(x)✳

❘❡➨✐t❡✈✳ ✭✭❛✮ ⇒ ✭❜✮✮ ◆❛❥ ❜♦ λ ∈ σ(ϕ(x)). P♦t❡♠ ❡❧❡♠❡♥t ϕ(x) − λ1 =

ϕ(x − λ1) ♥✐ ♦❜r♥❧❥✐✈✳ ●❧❡❞❡ ♥❛ ♣r❡❞♣♦st❛✈❦♦ x − λ1 ♥✐ ♦❜r♥❧❥✐✈ ❡❧❡♠❡♥t

✐♥ ③❛t♦ ❥❡ λ ∈ σ(x)✳

◆❛❥ ❜♦ λ ∈ σ(x)✳ P♦t❡♠ ❡❧❡♠❡♥t x − λ1 ♥✐ ♦❜r♥❧❥✐✈✳ P♦ ♣r❡❞♣♦st❛✈❦✐

t✉❞✐ ϕ(x − λ1) = ϕ(x) − λ1 ♥✐ ♦❜♥❧❥✐✈ ❡❧❡♠❡♥t✳ ■③ t❡❣❛ s❧❡❞✐ ➸❡❧❡♥♦✳

✭✭❜✮ ⇒ ✭❛✮✮ Pr❡❞♣♦st❛✈✐♠♦✱ ❞❛ x ♥✐ ♦❜r♥❧❥✐✈ ❡❧❡♠❡♥t✳ ❚♦r❡❥ ❥❡ 0 ∈

σ(x) = σ(ϕ(x))✱ ✐③ ↔❡s❛r s❧❡❞✐ ➸❡❧❡♥♦✳ P♦❞♦❜♥♦✱ ↔❡ ϕ(x) ♥✐ ♦❜r♥❧❥✐✈✱ ❥❡

0 ∈ σ(ϕ(x)) = σ(x) ✐♥ ③❛t♦ x ♥✐ ♦❜r♥❧❥✐✈ ❡❧❡♠❡♥t✳

✷✹✳ ◆❛❥ ❜♦ ♣r❡s❧✐❦❛✈❛ A : l2 → l2 ❞❡✜♥✐r❛♥❛ s ♣r❡❞♣✐s♦♠

A(x1, x2, x3, . . .) = (0, x1, x2, x3, . . .).

P♦❦❛➸✐t❡✱ ❞❛ ❥❡ A ∈ B(l2) ✐♥ ✐③r❛↔✉♥❛❥t❡ A ✱ A∗ t❡r σ(A)✳

❘❡➨✐t❡✈✳ ◆♦r♠❛ ♦♣❡r❛t♦r❥❛ A ❥❡ ✶✱ A∗(x1, x2, x3, x4, . . .) = (x2, x3, x4, . . .),

σ(A) = K(0, 1).

✷✺✳ ◆❛❥ ❜♦ ♣r❡s❧✐❦❛✈❛ A : l2 → l2 ❞❡✜♥✐r❛♥❛ s ♣r❡❞♣✐s♦♠

A(x1, x2, x3, . . .) = (x2, x3, . . .).

P♦❦❛➸✐t❡✱ ❞❛ ❥❡ A ∈ B(l2) ✐♥ ✐③r❛↔✉♥❛❥t❡ A ✱ A∗ t❡r σ(A)✳

❘❡➨✐t❡✈✳ ◆♦r♠❛ ♦♣❡r❛t♦r❥❛ A ❥❡ ✶✱ A∗(x1, x2, x3, . . .) = (0, x1, x2, x3, . . .),

σ(A) = K(0, 1).

✸✾

✷✻✳ ◆❛❥ ❜♦ ♣r❡s❧✐❦❛✈❛ U : l2 → l2 ❞❡✜♥✐r❛♥❛ s ♣r❡❞♣✐s♦♠

U (x1, x2, x3, x4, . . .) = (x1, −x2, x3, −x4, . . .).

P♦❦❛➸✐t❡✱ ❞❛ ❥❡ U ∈ B(l2) ✐♥ ✐③r❛↔✉♥❛❥t❡ U ✱ U∗ t❡r σ(U)✳

❘❡➨✐t❡✈✳ ◆♦r♠❛ ♦♣❡r❛t♦r❥❛ U ❥❡ ✶✱ U∗ = U✱ σ(U) = {1, −1}.

✷✼✳ ◆❛❥ ❜♦ ♣r❡s❧✐❦❛✈❛ T : l2 → l2 ❞❡✜♥✐r❛♥❛ s ♣r❡❞♣✐s♦♠

T (x1, x2, x3, x4, . . .) = (−x2, x1, −x4, x3, . . .).

P♦❦❛➸✐t❡✱ ❞❛ ❥❡ T ③✈❡③❡♥ ❧✐♥❡❛r❡♥ ♦♣❡r❛t♦r ✐♥ ✐③r❛↔✉♥❛❥t❡ T ✱ T ∗ t❡r

σ(T )✳

❘❡➨✐t❡✈✳ ◆♦r♠❛ ♦♣❡r❛t♦r❥❛ T ❥❡ ✶✱ T ∗ = −T ✱ σ(T ) = {i, −i}.

✷✽✳ ◆❛❥ ❜♦ ♣r❡s❧✐❦❛✈❛ A : l2 → l2 ❞❡✜♥✐r❛♥❛ s ♣r❡❞♣✐s♦♠

A(x1, x2, x3, x4, . . .) = (x1 + x2, x1 − x2, x3 + x4, x3 − x4, . . .).

P♦❦❛➸✐t❡✱ ❞❛ ❥❡ A ∈ B(l2) ✐♥ ✐③r❛↔✉♥❛❥t❡ A ✱ A∗ t❡r σ(A)✳

√

√

√

❘❡➨✐t❡✈✳ ◆♦r♠❛ ♦♣❡r❛t♦r❥❛ A ❥❡ 2✱ A∗ = A✱ σ(A) = { 2, − 2}.

✷✾✳ ◆❛❥ ❜♦ A : l2 → l2 ♦♣❡r❛t♦r ❞❡✜♥✐r❛♥ s ♣r❡❞♣✐s♦♠

A(x1, x2, x3, x4, . . .) = (x1 + 2x2, 2x1 − x2, x3 + 2x4, 2x3 − x4, . . .).

P♦❦❛➸✐t❡✱ ❞❛ ❥❡ A ∈ B(l2) ✐♥ ✐③r❛↔✉♥❛❥t❡ A ✱ A∗ t❡r σ(A)✳

√

√

√

❘❡➨✐t❡✈✳ ◆♦r♠❛ ♦♣❡r❛t♦r❥❛ A ❥❡ 5✱ A∗ = A✱ σ(A) = { 5, − 5}.

✹✵

✸✵✳ ◆❛❥ ❜♦st❛ a ✐♥ b ♥❡♥✐↔❡❧♥❛ ✈❡❦t♦r❥❛ ♥❡s❦♦♥↔♥♦ r❛③s❡➸♥❡❣❛ ❍✐❧❜❡rt♦✈❡❣❛

♣r♦st♦r❛ H ✐♥ ♥❛❥ ❜♦ A : H → H ❧✐♥❡❛r❡♥ ♦♣❡r❛t♦r ❞❡✜♥✐r❛♥ s ♣r❡❞♣✐s♦♠

Ax = x, a b✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ A ♦♠❡❥❡♥ ❧✐♥❡❛r❡♥ ♦♣❡r❛t♦r✳ ■③r❛↔✉♥❛❥t❡

♥❥❡❣♦✈♦ ♥♦r♠♦ t❡r A∗✳ ❑❞❛❥ ❥❡ A ♥♦r♠❛❧❡♥ ✐♥ ❦❞❛❥ s❡❜✐❛❞❥✉♥❣✐r❛♥ ♦♣✲

❡r❛t♦r❄ ❑❛❥ st❛ ❥❡❞r♦ ✐♥ ③❛❧♦❣❛ ✈r❡❞♥♦st✐ ♦♣❡r❛t♦r❥❛ A❄ ■③r❛↔✉♥❛❥t❡

σ(A)✳

◆❛♠✐❣✳ ◆♦r♠❛ ♦♣❡r❛t♦r❥❛ A ❥❡ a b ✱ ImA = {λb | λ ∈ C}✱ KerA =

{a}⊥✳ ❑❡r ❥❡

Ax, y =

x, a b, y = x, a b, y = x, y, b a = x, A∗y ,

❥❡ A∗x = x, b a✳ ❖♣❡r❛t♦r A ❥❡ ♥♦r♠❛❧❡♥✱ ❦♦ ❥❡ b = λa✱ λ ∈ C✳ ❖♣❡r❛t♦r

A ❥❡ s❡❜✐ ❛❞❥✉♥❣✐r❛♥✱ ❦♦ ❥❡ b = λa✱ λ ∈ R✳

◆❛❥ ❜♦ Ax = λx✱ x = 0✳ P♦t❡♠ st❛ 0 ✐♥ b, a ❧❛st♥✐ ✈r❡❞♥♦st✐ ♦♣❡r❛t♦r❥❛

A✳ ◆❛❥ ❜♦ λ = 0, b, a ✳ ■③ ❡♥❛❦♦st✐ (A − λI)x = y s❧❡❞✐ x, a b, a =

λ x, a + y, a ✳ ❙ ♣♦♠♦↔❥♦ t❡❣❛ ❧❛❤❦♦ ♣♦❦❛➸❡♠♦✱ ❞❛ ❥❡ x = 1 ( y,a b−y)✳

λ

b,a −λ

❚♦r❡❥ ❥❡ ♦♣❡r❛t♦r A − λI ♦❜r♥❧❥✐✈✳ ❙ t❡♠ s♠♦ ♣♦❦❛③❛❧✐✱ ❞❛ ❥❡ σ(A) =

{0, b, a }✳

✸✶✳ ◆❛❥ ❜♦ H ❍✐❧❜❡rt♦✈ ♣r♦st♦r✱ ❞✐♠H ≥ 4 ✐♥ ♥❛❥ ❜♦❞♦ a, b, c ∈ H ❧✐♥❡❛r♥♦

♥❡♦❞✈✐s♥✐ t❡r ♣❛r♦♠❛ ♦rt♦❣♦♥❛❧♥✐ ✈❡❦t♦r❥✐✳ ❉❡✜♥✐r❛❥♠♦ ♦♣❡r❛t♦r A :

H → H s ♣r❡❞♣✐s♦♠

Ax = x, a a + x, b b + x, c c.

P♦❦❛➸✐t❡✱ ❞❛ ❥❡ A ∈ B(H) ✐♥ ✐③r❛↔✉♥❛❥t❡ σ(A) t❡r A∗✳ ❆❧✐ ❥❡ A ✐♥❥❡❦t✐✈❡♥

♦♣❡r❛t♦r❄ ❆❧✐ ❥❡ s✉r❥❡❦t✐✈❡♥❄

◆❛♠✐❣✳ ▲❛st♥❡ ✈r❡❞♥♦st✐ ♦♣❡r❛t♦r❥❛ A s♦ 0, a 2, b 2, c 2✳ ❷❡ λ ♥✐

❧❛st♥❛ ✈r❡❞♥♦st ♦♣❡r❛t♦r❥❛ A✱ ♣♦t❡♠ ✐③ ❡♥❛❦♦st✐ (A − λI)x = y s❧❡❞✐

x, a =

y,a

✱ x, b = y,b ✐♥ x, c = y,c . ❚♦r❡❥ ❥❡

a 2−λ

b 2−λ

c 2−λ

1

x =

( x, a a + x, b b + x, c c − y),

λ

❦❛r ♣♦♠❡♥✐✱ ❞❛ ❥❡ ♦♣❡r❛t♦r A − λI ♦❜r♥❧❥✐✈✳ ■③ t❡❣❛ s❧❡❞✐✱ ❞❛ ❥❡ σ(A) =

{0, a 2, b 2, c 2}✳

❖♣❡r❛t♦r A ♥✐ ♥✐t✐ ✐♥❥❡❦t✐✈❡♥ ♥✐t✐ s✉r❥❡❦t✐✈❡♥✱ s❛❥ ❥❡ ❞✐♠H ≥ 4✱ A = A∗✳

✹✶

✸✷✳ ◆❛❥ ❜♦ H ❦♦♠♣❧❡❦s❡♥ ❍✐❧❜❡rt♦✈ ♣r♦st♦r ✐♥ A ∈ B(H) t❛❦ ♦♣❡r❛t♦r✱ ❞❛

❥❡ A∗ = −A✳

✭❛✮ P♦❦❛➸✐t❡✱ ❞❛ ✐③ A2 = 0 s❧❡❞✐ A = 0✳

✭❜✮ ◆❛❥ ❜♦ λ ❧❛st♥❛ ✈r❡❞♥♦st ♦♣❡r❛t♦r❥❛ A✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ λ = it ③❛

♥❡❦ t ∈ [− A , A ]✳

◆❛♠✐❣✳

✭❛✮ ■③ ❡♥❛❦♦st✐ 0 = x, A2x = A∗x, Ax s❧❡❞✐ ➸❡❧❡♥♦✳

✭❜✮ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ λ x, x = −λ x, x ✱ ✐③ ↔❡s❛r s❧❡❞✐ ➸❡❧❡♥♦✳

✸✸✳ ◆❛❥ ❜♦ H ❍✐❧❜❡rt♦✈ ♣r♦st♦r ✐♥ ♥❛❥ ❜♦ A ∈ B(H) t❛❦ ♥♦r♠❛❧❡♥ ♦♣❡r❛t♦r✱

❞❛ ❥❡ A2 = −A✳ P♦❦❛➸✐t❡✿

✭❛✮ A∗Ax, A∗y = − A∗x, A∗y ✱ x, y ∈ H✱

✭❜✮ A = A∗✱

✭❝✮ {0, −1} ⊆ σ(A)✳

❘❡➨✐t❡✈✳

✭❛✮ ❱✐❞✐♠♦✱ ❞❛ ❥❡

A∗Ax, A∗y = AA∗x, A∗y = A∗x, (A2)∗y = − A∗x, A∗y

③❛ ✈s❛❦ x, y ∈ H✳

✭❜✮ ●❧❡❞❡ ♥❛ ③❣♦r♥❥♦ ❡♥❛❦♦st ♦♣❛③✐♠♦✱ ❞❛ ❥❡

0 = (A∗A + A∗)x, A∗y = A∗(A + I)x, A∗y

③❛ ✈s❛❦ x, y ∈ H✳ P✐➨✐♠♦ (A + I)x ♥❛♠❡st♦ y✳ P♦t❡♠ ❥❡ A∗(A +

I)x = 0 ③❛ ✈s❛❦ x ∈ H✱ ❦❛r ♥❛s ♣r✐✈❡❞❡ ❞♦ ➸❡❧❡♥❡ ❡♥❛❦♦st✐✱ s❛❥ ❥❡

0 = A∗A + A∗ = A∗A + A✳

✭❝✮ ◆❛❥ ❜♦ λ ❧❛st♥❛ ✈r❡❞♥♦st ♦♣❡r❛t♦r❥❛ A✳ P♦t❡♠ ♦❜st❛❥❛ t❛❦ 0 = x ∈

H✱ ❞❛ ❥❡ Ax = λx✳ ❖♣❛③✐♠♦✱ ❞❛ ❥❡ λ x, x = λ x, x ✳ ❯♣♦➨t❡✈❛❥♠♦

♣r❡❞♣♦st❛✈❦♦✱ ❞❛ ❥❡ A2 = −A✱ ❦❛r ♥❛s ♣r✐✈❡❞❡ ❞♦ t❡❣❛✱ ❞❛ ❥❡ λ = 0

❛❧✐ λ = −1✳

✹✷

✸✹✳ ◆❛❥ ❜♦ H ❍✐❧❜❡rt♦✈ ♣r♦st♦r ✐♥ ♥❛❥ ❜♦ A ∈ B(H) t❛❦ ♦♣❡r❛t♦r✱ ❞❛ ❥❡

A3 = A∗✳

✭❛✮ P♦❦❛➸✐t❡✱ ❞❛ ✐③ A4 = 0 s❧❡❞✐ A = 0✳

✭❜✮ P♦✐➨↔✐t❡ ❧❛st♥❡ ✈r❡❞♥♦st✐ ♦♣❡r❛t♦r❥❛ A✳

✭❝✮ ❉♦❧♦↔✐t❡ s♣❡❦t❡r ♦♣❡r❛t♦r❥❛ A✱ ↔❡ ❥❡ A4 = I✳

❘❡➨✐t❡✈✳

✭❛✮ ❑❡r ❥❡ Ax, Ax = x, A4x = 0✱ ❥❡ Ax = 0 ③❛ ✈s❛❦ x ∈ H✳

✭❜✮ ▲❛st♥❡ ✈r❡❞♥♦st✐ ♦♣❡r❛t♦r❥❛ A s♦ 0, 1, −1, i, −i✳

✭❝✮ ❙♣❡❦t❡r ♦♣❡r❛t♦r❥❛ A ❥❡ σ(A) = {1, −1, i, −i}✳

✸✺✳ ◆❛❥ ❜♦ H ❍✐❧❜❡rt♦✈ ♣r♦st♦r ✐♥ ♥❛❥ ❜♦ A ∈ B(H) ♣♦③✐t✐✈❡♥ ♦♣❡r❛t♦r✳

P♦❦❛➸✐t❡✿

✭❛✮ s✉♣{ Ax, x | x = 1} ≤ 1 ♥❛t❛♥❦♦ t❡❞❛❥✱ ❦♦ ❥❡ Ax, x ≤ x, x ✱

✭❜✮ ↔❡ ❥❡ λ ❧❛st♥❛ ✈r❡❞♥♦st ♦♣❡r❛t♦r❥❛ A✱ ❥❡ 0 ≤ λ.

❘❡➨✐t❡✈✳

✭❛✮ ✭=⇒✮ ❷❡ ❥❡ x = 1✱ ♦↔✐t♥♦ s❧❡❞✐ ➸❡❧❡♥♦✳ ◆❛❥ ❜♦ x = 1✳ P♦t❡♠

❥❡ x , x = 1. ❙❧❡❞✐ A x , x ≤ 1. ❚♦r❡❥ ❥❡ Ax, x ≤ x, x .

x

x

x

x

✭⇐=✮ P♦ ♣r❡❞♣♦st❛✈❦✐ ❥❡ Ax, x ≤ x, x ✳ ❷❡ ❥❡ x = 1✱ s❧❡❞✐

➸❡❧❡♥♦✳

✭❜✮ ❑❡r ❥❡ λ ❧❛st♥❛ ✈r❡❞♥♦st ♦♣❡r❛t♦r❥❛ A✱ ♦❜st❛❥❛ t❛❦ 0 = x ∈ H✱ ❞❛

❥❡ Ax = λx✳ P♦t❡♠ ❥❡ Ax, x = λ x, x ✳ P♦ ❞r✉❣✐ str❛♥✐ ♣❛ ❥❡

x, Ax = λ x, x ✳ ■③ t❡❣❛ s❧❡❞✐ ➸❡❧❡♥♦✳

✸✻✳ ◆❛❥ ❜♦ H ❍✐❧❜❡rt♦✈ ♣r♦st♦r ✐♥ A ∈ B(H)✳ Pr❡❞♣♦st❛✈✐♠♦✱ ❞❛ ❥❡ AA∗

❦♦♠♣❛❦t❡♥ ♦♣❡r❛t♦r✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ ♣♦t❡♠ ❦♦♠♣❛❦t❡♥ t✉❞✐ ♦♣❡r❛t♦r A✳

✹✸

◆❛♠✐❣✳ ◆❛❥ ❜♦ {xn}n∈N ♦♠❡❥❡♥♦ ③❛♣♦r❡❞❥❡ ✈ H✳ ❑❡r ❥❡ AA∗ ❦♦♠♣❛❦✲

t❡♥ ♦♣❡r❛t♦r✱ ✈s❡❜✉❥❡ ③❛♣♦r❡❞❥❡ {A∗Axn}n∈N ❦♦♥✈❡r❣❡♥t♥♦ ♣♦❞③❛♣♦r❡❞❥❡

{A∗Axn }

p

p∈N✳ ❙ ♣♦♠♦↔❥♦ ♥❡❡♥❛❦♦st✐

Ax 2 = Ax, Ax = A∗Ax, x ≤ A∗Ax x

♦♣❛③✐♠♦✱ ❞❛ ❥❡ ③❛♣♦r❡❞❥❡ {Axn }

p

p∈N ❈❛✉❝❤②❥❡✈♦ ✐♥ ③❛t♦ ❦♦♥✈❡r❣❡♥t♥♦✳

✸✼✳ ◆❛❥ ❜♦ H ❍✐❧❜❡rt♦✈ ♣r♦st♦r ✐♥ A ∈ B(H)✳

✭❛✮ P♦❦❛➸✐t❡✱ ❞❛ ③❛ ✈s❛❦ x ∈ H ✈❡❧❥❛

Ax 4 ≤ x 3 (A∗A)2x .

✭❜✮ ◆❛❥ ❜♦ (A∗A)2 ❦♦♠♣❛❦t❡♥ ♦♣❡r❛t♦r✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ ♣♦t❡♠ ❦♦♠✲

♣❛❦t❡♥ t✉❞✐ ♦♣❡r❛t♦r A✳

◆❛♠✐❣✳

✭❛✮ ❖↔✐t♥♦ ❥❡

Ax 4 = ( x, A∗Ax )2 ≤ x 2 A∗Ax 2

=

x 2 x, (A∗A)2x ≤ x 3 (A∗A)2x .

✭❜✮ ◆❛❥ ❜♦ {xn}n∈N ♦♠❡❥❡♥♦ ③❛♣♦r❡❞❥❡ ✈ H✳ ❙ ♣♦♠♦↔❥♦ ♥❡❡♥❛❦♦st✐ ✭❛✮

✐♥ ♣r❡❞♣♦st❛✈❦❡ ♣♦❦❛➸❡♠♦✱ ❞❛ ✈s❡❜✉❥❡ ③❛♣♦r❡❞❥❡ {(A∗A)2xn}n∈N

❦♦♥✈❡r❣❡♥t♥♦ ♣♦❞③❛♣♦r❡❞❥❡ {(A∗A)2xn }

p

p∈N✳

■③ t❡❣❛ s❧❡❞✐ ✭❣❧❡❥t❡

t♦↔❦♦ ✭❛✮✮✱ ❞❛ ❥❡ t✉❞✐ ③❛♣♦r❡❞❥❡ {Axn }

p

p∈N ❦♦♥✈❡r❣❡♥t♥♦✳

✸✽✳ ◆❛❥ ❜♦st❛ X ✐♥ Y ❇❛♥❛❝❤♦✈❛ ♣r♦st♦r❛ ✐♥ A : X → Y ❦♦♠♣❛❦t❡♥ ❧✐♥❡❛r❡♥

♦♣❡r❛t♦r✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ ImA ③❛♣rt ♣♦❞♣r♦st♦r Y ♥❛t❛♥❦♦ t❡❞❛❥✱ ❦♦ ❥❡

A ❦♦♥↔♥❡❣❛ r❛♥❣❛✳

◆❛♠✐❣✳ ✭⇐=) ❱s❛❦ ❦♦♥↔♥♦ r❛③s❡➸❡♥ ♣♦❞♣r♦st♦r ♥♦r♠✐r❛♥❡❣❛ ♣r♦st♦r❛

❥❡ ③❛♣rt✳ ❷❡ ❥❡ t♦r❡❥ ♦♣❡r❛t♦r A ❦♦♥↔♥❡❣❛ r❛♥❣❛✱ ♣♦t❡♠ ❥❡ ImA ③❛♣rt

♣♦❞♣r♦st♦r ♣r♦st♦r❛ Y ✳

✹✹

✭=⇒✮ ◆❛❥ ❜♦ A : X → Y t❛❦ ❦♦♠♣❛❦t❡♥ ❧✐♥❡❛r❡♥ ♦♣❡r❛t♦r✱ ❞❛ ❥❡ ImA

③❛♣rt ♣♦❞♣r♦st♦r ♣r♦st♦r❛ Y ✳ ▲✐♥❡❛r❡♥ ♦♣❡r❛t♦r A : X → ImA ❥❡ s✉r✲

❥❡❦t✐✈❡♥ ✐♥ ♦♠❡❥❡♥✳ P♦ ✐③r❡❦✉ ♦ ♦❞♣rt✐ ♣r❡s❧✐❦❛✈✐ ❥❡ A(K(0, 1))✱ ❦❥❡r

K(0, 1) ♦③♥❛↔✉❥❡ ♦❞♣rt♦ ❡♥♦ts❦♦ ❦r♦❣❧♦ s sr❡❞✐➨↔❡♠ ✈ 0✱ ♦❞♣rt❛ ♣♦❞♠✲

♥♦➸✐❝❛ ImA✳ ❚♦r❡❥ ♦❜st❛❥❛ ♥❡❦❛ ③❛♣rt❛ ❦r♦❣❧❛ K(0, ǫ) ⊆ A(K(0, 1))✳ ❑❡r

❥❡ A(K(0, 1)) ❦♦♠♣❛❦t♥❛ ♠♥♦➸✐❝❛ ✐♥ ❥❡ ③❛♣rt ♣♦❞♣r♦st♦r ❦♦♠♣❛❦t♥❡❣❛

♣r♦st♦r❛ ❦♦♠♣❛❦t❡♥✱ ❥❡ K(0, ǫ) ❦♦♠♣❛❦t♥❛ ♣♦❞♠♥♦➸✐❝❛ ImA ✐♥ ❥❡ ③❛t♦

dim(ImA) < ∞✳

✸✾✳ ◆❛❥ ❜♦st❛ X ✐♥ Y ❇❛♥❛❝❤♦✈❛ ♣r♦st♦r❛ t❡r ♥❛❥ ❜♦ A : X → Y ③✈❡③❡♥✱

s✉r❥❡❦t✐✈❡♥ ♦♣❡r❛t♦r✳ P♦❦❛➸✐t❡✿ ♦❜st❛❥❛ t❛❦ m > 0✱ ❞❛ ③❛ ✈s❛❦ y ∈ Y

♦❜st❛❥❛ t❛❦ x ∈ X✱ ❞❛ ❥❡ Ax = y ✐♥ x ≤ m Ax ✳

❘❡➨✐t❡✈✳ ◆❛❥ ❜♦ K(0, 1) = {x ∈ X | x < 1}✳ P♦ ✐③r❡❦✉ ♦ ♦❞♣rt✐

♣r❡s❧✐❦❛✈✐ ♦♣❡r❛t♦r A s❧✐❦❛ ♦❞♣rt❡ ♠♥♦➸✐❝❡ ✈ ♦❞♣rt❡ ♠♥♦➸✐❝❡✳ P♦t❡♠

♦❜st❛❥❛ t❛❦❛ ♦❞♣rt❛ ♠♥♦➸✐❝❛ K(0, r)✱ ❞❛ ❥❡ K(0, r) ⊆ A(K(0, 1))✳ ◆❛❥ ❜♦

0 = y ∈ Y ✐♥ ♣✐➨✐♠♦ z = yr ✳ ❚♦r❡❥ ❥❡ z = r ✐♥ ③❛t♦ ❥❡ z ∈ A(K(0, 1))✳

y 2

2

■③ t❡❣❛ s❧❡❞✐✱ ❞❛ ♦❜st❛❥❛ t❛❦ x0 ∈ K(0, 1)✱ ❞❛ ❥❡ z = Ax0✳ ◆❛❥ ❜♦

x = x y 2

0

✳ P♦t❡♠ ❥❡ Ax = y ✐♥ x = 2 y x

Ax ✳

r

r

0

≤ 2r

✹✵✳ ◆❛❥ ❜♦ ✈❡❦t♦rs❦✐ ♣r♦st♦r X ❇❛♥❛❝❤♦✈ ③❛ ♥♦r♠✐ . 1 ✐♥ . 2✳ Pr❡❞✲

♣♦st❛✈✐♠♦✱ ❞❛ ❥❡ x 1 ≤ x 2 ③❛ ✈s❛❦ x ∈ X✳ P♦❦❛➸✐t❡✱ ❞❛ st❛ t❡❞❛❥

♥♦r♠✐ ❡❦✈✐✈❛❧❡♥t♥✐✳

◆❛♠✐❣✳ ❑❡r ❥❡ id : (X, . 2) → (X, . 1) ❧✐♥❡❛r♥❛✱ ③✈❡③♥❛ ✐♥ ❜✐❥❡❦t✐✈♥❛

♣r❡s❧✐❦❛✈❛ ✭✉♣♦➨t❡✈❛❥t❡✱ ❞❛ ❥❡ id(x) 1 = x 1 ≤ x 2 ③❛ ✈s❛❦ x ∈ X✮✱

❥❡ ♣♦ ✐③r❡❦✉ ♦ ♦❞♣rt✐ ♣r❡s❧✐❦❛✈✐ id−1 ③✈❡③♥❛ ♣r❡s❧✐❦❛✈❛✳ ❙❧❡❞✐ x 2 =

id−1(x) 2 ≤ m x 1✳

✹✶✳ ◆❛❥ ❜♦ H ❍✐❧❜❡rt♦✈ ♣r♦st♦r ✐♥ T ∈ B(H)✳ P♦❦❛➸✐t❡✿

✭❛✮ KerT ∗ = (ImT )⊥✱

✭❜✮ (I + T ∗T )x ≥ x ✱

✭❝✮ I + T ∗T ❥❡ ♦❜r♥❧❥✐✈ ♦♣❡r❛t♦r ✈ B(H)✳

✹✺

❘❡➨✐t❡✈✳

✭❛✮ ◆❛❥ ❜♦ x ∈ KerT ∗✳ P♦t❡♠ ❥❡ 0 = T ∗x, y = x, T y ③❛ ✈s❛❦ y ∈ H✳

■③ t❡❣❛ s❧❡❞✐✱ ❞❛ ❥❡ x ∈ (ImT )⊥✳

◆❛❥ ❜♦ x ∈ (ImT )⊥✳ P♦t❡♠ ❥❡ 0 = x, T y = T ∗x, y ③❛ ✈s❛❦

y ∈ H✳ ❚♦r❡❥ ❥❡ T ∗x = 0 ✐♥ ③❛t♦ ❥❡ x ∈ KerT ∗✳

✭❜✮ ❱✐❞✐♠♦✱ ❞❛ ❥❡

(I+T ∗T )x 2 = (I+T ∗T )x, (I+T ∗T )x = x 2+ T ∗T x 2+2 T x 2,

✐③ ↔❡s❛r s❧❡❞✐ ➸❡❧❡♥♦✳

✭❝✮ ❙ ♣♦♠♦↔❥♦ t♦↔❦❡ ✭❜✮ ♦♣❛③✐♠♦✱ ❞❛ ❥❡ ♦♣❡r❛t♦r I + T ∗T ✐♥❥❡❦t✐✈❡♥✱ s

♣♦♠♦↔❥♦ t♦↔❦❡ ✭❛✮ ♣❛ ♣♦❦❛➸❡♠♦ s✉r❥❡❦t✐✈♥♦st ♦♣❡r❛t♦r❥❛ I + T ∗T ✳

❯♣♦➨t❡✈❛❥t❡✱ ❞❛ ❥❡

H = Im(I + T ∗T ) ⊕ (Im(I + T ∗T ))⊥,

Ker(I + T ∗T ) = (Im(I + T ∗T ))⊥ = {0}.

P♦ ✐③r❡❦✉ ♦ ♦❞♣rt✐ ♣r❡s❧✐❦❛✈✐ s❧❡❞✐✱ ❞❛ ❥❡ I + T ∗T ♦❜r♥❧❥✐✈ ♦♣❡r❛t♦r✳

✹✷✳ ◆❛❥ ❜♦ X ❇❛♥❛❝❤♦✈ ♣r♦st♦r ✐♥ ♥❛❥ ❜♦st❛ Y t❡r Z t❛❦❛ ③❛♣rt❛ ♣♦❞♣r♦st♦r❛

X✱ ❞❛ ❥❡ X = Y ⊕ Z✳ P♦❦❛➸✐t❡✱ ❞❛ st❛ ♣r♦st♦r❛ X/Y ✐♥ Z ❧✐♥❡❛r♥♦

❤♦♠❡♦♠♦r❢♥❛✳

❘❡➨✐t❡✈✳ ◆❛❥♣r❡❥ ③❛♣✐➨✐♠♦ ✐③r❡❦✿

❷❡ ❥❡ Y ③❛♣rt ♣♦❞♣r♦st♦r ♥♦r♠✐r❛♥❡❣❛ ♣r♦st♦r❛ X✱ ♣♦t❡♠ ❥❡ s ♣r❡❞♣✐s♦♠

x + Y = infy∈Y x + y

❞❡✜♥✐r❛♥❛ ♥♦r♠❛ ♥❛ X/Y ✳ ❷❡ ❥❡ X ❇❛♥❛❝❤♦✈ ♣r♦st♦r✱ ♣♦t❡♠ ❥❡ t✉❞✐

X/Y ❇❛♥❛❝❤♦✈ ♣r♦st♦r✳

❉❡✜♥✐r❛❥♠♦ ♣r❡s❧✐❦❛✈♦ ϕ : Z → X/Y s ♣r❡❞♣✐s♦♠ ϕ(z) = z + Y ✳ ◆❛❥

❜♦ z ∈ Kerϕ✳ P♦t❡♠ ❥❡ z ∈ Y ∩ Z = {0}. ❙ t❡♠ s♠♦ ♣♦❦❛③❛❧✐✱ ❞❛ ❥❡

♣r❡s❧✐❦❛✈❛ ϕ ✐♥❥❡❦t✐✈♥❛✳ Pr❛✈ t❛❦♦ ♥✐ t❡➸❦♦ ♣r❡✈❡r✐t✐✱ ❞❛ ❥❡ s✉r❥❡❦t✐✈♥❛✳

◆❛♠r❡↔✱ ♥❛❥ ❜♦ x + Y ∈ X/Y ✳ ❑❡r ❥❡ X = Y ⊕ Z✱ ❥❡ x = y + z ③❛ ♥❡❦❛

y ∈ Y ✱ z ∈ Z✳ ■③ t❡❣❛ s❧❡❞✐✱ ❞❛ ❥❡ x − z ∈ Y ✐♥ ③❛t♦ ❥❡ x + Y = z + Y ✳

✹✻

❚♦r❡❥ ♦❜st❛❥❛ t❛❦ z ∈ Z✱ ❞❛ ❥❡ ϕ(z) = x + Y ✳ ◆❛ ❦♦♥❝✉ ➨❡ ♣r❡✈❡r✐♠♦✱ ❞❛

❥❡ ϕ ≤ 1✿

ϕ(z) = infy∈Y z + y ≤ infy∈Y ( z + y ) ≤ z .

❚♦r❡❥ ❥❡ ϕ ③✈❡③♥❛ ♣r❡s❧✐❦❛✈❛✳ P♦ ✐③r❡❦✉ ♦ ♦❞♣rt✐ ♣r❡s❧✐❦❛✈✐ ❥❡ ϕ−1 ③✈❡③♥❛

♣r❡s❧✐❦❛✈❛✳

✹✸✳ ◆❛❥ ❜♦ X ❇❛♥❛❝❤♦✈ ♣r♦st♦r ✐♥ P ♣r♦❥❡❦t♦r ♥❛ X✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ P

③✈❡③❡♥ ♦♣❡r❛t♦r ♥❛t❛♥❦♦ t❡❞❛❥✱ ❦♦ st❛ KerP ✐♥ ImP ③❛♣rt❛ ♣♦❞♣r♦st♦r❛

X✳

❘❡➨✐t❡✈✳ ✭⇐=✮ Pr❡❞♣♦st❛✈✐♠♦✱ ❞❛ st❛ KerP ✐♥ ImP ③❛♣rt❛ ♣♦❞♣r♦s✲

t♦r❛ X✳ ◆❛❥ ❜♦ {xn}n∈N ③❛♣♦r❡❞❥❡ ✈ X ③ ❧✐♠✐t♦ 0 ✐♥ {Pxn}n∈N ③❛♣♦r❡❞❥❡

✈ X ③ ❧✐♠✐t♦ y✳ P♦ ✐③r❡❦✉ ♦ ③❛♣rt❡♠ ❣r❛❢✉ ③❛❞♦➨↔❛ ♣♦❦❛③❛t✐✱ ❞❛ ❥❡ y = 0✳

❑❡r ❥❡ P (P xn − xn) = 0✱ ❥❡ P xn − xn ∈ KerP ✳ ●❧❡❞❡ ♥❛ ♣r❡❞♣♦st❛✈❦♦

s❧❡❞✐✱ ❞❛ ❥❡ y ∈ KerP ✳ P♦t❡♠ ❥❡ 0 = P y = y✳

✭=⇒✮ Pr❡❞♣♦st❛✈✐♠♦✱ ❞❛ ❥❡ P ③✈❡③❡♥ ♣r♦❥❡❦t♦r✳ ◆❛❥ ❜♦ {xn}n∈N ⊂ X

③❛♣♦r❡❞❥❡ ✈ KerP ③ ❧✐♠✐t♦ x✳ P♦t❡♠ ❥❡ x ∈ KerP ✳ ◆❛♠r❡↔✱

P x = P lim xn = lim P xn = 0.

n→∞

n→∞

Pr❛✈ t❛❦♦ ❥❡ ImP ③❛♣rt ♣♦❞♣r♦st♦r ♣r♦st♦r❛ X✱ s❛❥ ❥❡ Ker(P − I) =

ImP ✳

✹✹✳ ◆❛❥ ❜♦st❛ P ✐♥ Q t❛❦❛ ♣r♦❥❡❦t♦r❥❛ ♥❛ ❇❛♥❛❝❤♦✈❡♠ ♣r♦st♦r✉ X✱ ❞❛ ❥❡

P Q = −QP ✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ P + Q ③✈❡③❡♥ ♦♣❡r❛t♦r ♥❛t❛♥❦♦ t❡❞❛❥✱ ❦♦

st❛ Im(P + Q) ✐♥ Ker(P + Q) ③❛♣rt❛ ♣♦❞♣r♦st♦r❛ ♣r♦st♦r❛ X✳

◆❛♠✐❣✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ P + Q ♣r♦❥❡❦t♦r ✐♥ ✉♣♦➨t❡✈❛❥t❡ ♣r❡❥➨♥❥♦ ♥❛❧♦❣♦✳

✹✺✳ ◆❛❥ ❜♦ X ❇❛♥❛❝❤♦✈ ♣r♦st♦r✱ A : X → X ❧✐♥❡❛r❡♥ ♦♣❡r❛t♦r ✐♥ ♥❛❥ ❜♦

B ∈ B(X) t❛❦ ❜✐❥❡❦t✐✈❡♥ ♦♣❡r❛t♦r✱ ❞❛ ❥❡ BA ∈ B(X)✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡

♣♦t❡♠ t✉❞✐ A ∈ B(X)✳

✹✼

◆❛♠✐❣✳ ●❧❡❞❡ ♥❛ ♣r❡❞♣♦st❛✈❦♦✱ ❞❛ ❥❡ B ∈ B(X) ❜✐❥❡❦t✐✈❡♥ ♦♣❡r❛t♦r✱

♦❜st❛❥❛ B−1 ∈ B(X)✳ ■③ t❡❣❛ s❧❡❞✐✱ ❞❛ ❥❡ A = B−1BA ∈ B(X)✳ ◆❛❧♦❣♦

❧❛❤❦♦ r❡➨✐♠♦ t✉❞✐ s ♣♦♠♦↔❥♦ ✐③r❡❦❛ ♦ ③❛♣rt❡♠ ❣r❛❢✉✳

✹✻✳ ◆❛❥ ❜♦ X ❇❛♥❛❝❤♦✈ ♣r♦st♦r✱ A : X → X ❧✐♥❡❛r❡♥ ♦♣❡r❛t♦r ✐♥ ♥❛❥ ❜♦

B ∈ B(X) t❛❦ ✐♥❥❡❦t✐✈❡♥ ♦♣❡r❛t♦r✱ ❞❛ ❥❡ BA ∈ B(X)✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡

♣♦t❡♠ t✉❞✐ A ∈ B(X)✳

❘❡➨✐t❡✈✳ ◆❛❥ ❜♦ {xn}n∈N ⊂ X ③❛♣♦r❡❞❥❡ ③ ❧✐♠✐t♦ 0 ✐♥ {Axn}n∈N ⊂ X

③❛♣♦r❡❞❥❡ ③ ❧✐♠✐t♦ y✳ P♦ ✐③r❡❦✉ ♦ ③❛♣rt❡♠ ❣r❛❢✉ ③❛❞♦➨↔❛ ♣♦❦❛③❛t✐✱ ❞❛

❥❡ y = 0✳ ❑❡r ❥❡ B ✐♥❥❡❦t✐✈❡♥ ♦♣❡r❛t♦r ✐♥ ❥❡ By = B(limn→∞ Axn) =

limn→∞ B(Axn) = BA(limn→∞ xn) = 0✱ ❥❡ y = 0✳

✹✼✳ ◆❛❥ ❜♦ X ❇❛♥❛❝❤♦✈ ♣r♦st♦r ✐♥ A : X → X ❧✐♥❡❛r❡♥ ♦♣❡r❛t♦r✳ Pr❡❞✲

♣♦st❛✈✐♠♦✱ ❞❛ ③❛ ✈s❛❦ f ∈ X∗ ✈❡❧❥❛ f ◦ A ∈ X∗✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡

A ∈ B(X)✳

❘❡➨✐t❡✈✳ ◆❛❥ ❜♦ {xn}n∈N ⊂ X ③❛♣♦r❡❞❥❡ ③ ❧✐♠✐t♦ 0 ✐♥ {Axn}n∈N ⊂ X ③❛✲

♣♦r❡❞❥❡ ③ ❧✐♠✐t♦ y✳ P♦ ✐③r❡❦✉ ♦ ③❛♣rt❡♠ ❣r❛❢✉ ③❛❞♦➨↔❛ ♣♦❦❛③❛t✐✱ ❞❛ ❥❡ y =

0✳ Pr❡❞♣♦st❛✈✐♠♦ ♥❛s♣r♦t♥♦✳ P♦t❡♠ ♣♦ ♣♦s❧❡❞✐❝✐ ❍❛❤♥ ❇❛♥❛❝❤♦✈❡❣❛

✐③r❡❦❛ ♦❜st❛❥❛ t❛❦ ❧✐♥❡❛r❡♥ ❢✉♥❦❝✐♦♥❛❧ f ∈ X∗✱ ❞❛ ❥❡ f(y) = 0✳ ❑❡r ❥❡

f ◦ A ∈ X∗ ✐♥ ③❛♣♦r❡❞❥❡ {xn}n∈N ❦♦♥✈❡r❣✐r❛ ❦ 0✱ ❥❡ limn→∞ f(Axn) = 0✳

■③ t❡❣❛ s❧❡❞✐✱ ❞❛ ❥❡ f(y) = f(limn→∞ Axn) = limn→∞ f(Axn) = 0✳ ❙ t❡♠

s♠♦ ♣r✐➨❧✐ ❞♦ ♣r♦t✐s❧♦✈❥❛✳ ❚♦r❡❥ ❥❡ y = 0 ✐♥ tr❞✐t❡✈ ❥❡ ❞♦❦❛③❛♥❛✳

✹✽✳ ◆❛❥ ❜♦ X ❇❛♥❛❝❤♦✈ ♣r♦st♦r ✐♥ ♥❛❥ ❜♦ F ♣♦❞♠♥♦➸✐❝❛ B(X) ③ ❧❛st♥♦st❥♦✿

③❛ ✈s❛❦ 0 = x ∈ X ♦❜st❛❥❛ t❛❦ A ∈ F✱ ❞❛ ❥❡ Ax = 0✳ ◆❛❥ ❜♦ B : X → X

t❛❦ ❧✐♥❡❛r❡♥ ♦♣❡r❛t♦r✱ ❞❛ ❥❡ f ◦ AB ∈ X∗ ③❛ ✈s❛❦ f ∈ X∗ ✐♥ A ∈ F✳

P♦❦❛➸✐t❡✱ ❞❛ ❥❡ B ∈ B(X)✳

◆❛♠✐❣✳ ●❧❡❞❡ ♥❛ ♣r❡❥➨♥❥♦ ♥❛❧♦❣♦ s❧❡❞✐✱ ❞❛ ❥❡ AB ∈ B(X)✳ ❙ ♣♦♠♦↔❥♦

✐③r❡❦❛ ♦ ③❛♣rt❡♠ ❣r❛❢✉ ♣♦❦❛➸✐t❡✱ ❞❛ ❥❡ B ∈ B(X)✳

✹✾✳ ◆❛❥ ❜♦ X ❇❛♥❛❝❤♦✈ ♣r♦st♦r ✐♥ ♥❛❥ ❜♦ 0 = u ∈ X✳

✹✽

✭❛✮ ❩❛ ✈s❛❦ f ∈ X∗ ♥❛❥ ❜♦ Tf : X → X ♦♣❡r❛t♦r ❞❡✜♥✐r❛♥ s ♣r❡❞♣✐s♦♠

Tf x = f (x)u✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ Tf ∈ B(X)✳

✭❜✮ ◆❛❥ ❜♦ A : X → X t❛❦ ❧✐♥❡❛r❡♥ ♦♣❡r❛t♦r✱ ❞❛ ❥❡ TfA ∈ B(X) ③❛

✈s❛❦ f ∈ X∗✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ A ∈ B(X)✳

◆❛♠✐❣✳ ✭❜✮ ◆❛❥ ❜♦ {xn}n∈N ⊂ X ③❛♣♦r❡❞❥❡ ③ ❧✐♠✐t♦ 0 ✐♥ {Axn}n∈N ⊂ X

③❛♣♦r❡❞❥❡ ③ ❧✐♠✐t♦ y✳ P♦ ✐③r❡❦✉ ♦ ③❛♣rt❡♠ ❣r❛❢✉ ③❛❞♦➨↔❛ ♣♦❦❛③❛t✐✱ ❞❛ ❥❡

y = 0✳ ❑❡r ❥❡ Tf A ∈ B(X) ③❛ ✈s❛❦ f ∈ X∗✱ ♥✐ t❡➸❦♦ ✈✐❞❡t✐✱ ❞❛ ❥❡

0 = Tf (y) = f (y)u ③❛ ✈s❛❦ f ∈ X∗✱ ✐③ ↔❡s❛r s❧❡❞✐ ➸❡❧❡♥♦✳

✺✵✳ ◆❛❥ ❜♦ X ❇❛♥❛❝❤♦✈ ♣r♦st♦r ✐♥ ♥❛❥ ❜♦ A : X → X t❛❦ ❧✐♥❡❛r❡♥ ♦♣❡r❛t♦r✱

❞❛ ❥❡ AT − T A ∈ B(X) ③❛ ✈s❛❦ T ∈ B(X)✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ A ∈ B(X)✳

❘❡➨✐t❡✈✳ ◆❛❥ ❜♦ 0 = u ∈ X t❛❦✱ ❞❛ ❥❡ u = 1✳ ❩❛ ✈s❛❦ f ∈ X∗

❞❡✜♥✐r❛❥♠♦ ♦♣❡r❛t♦r Tf : X → X s ♣r❡❞♣✐s♦♠ Tfx = f(x)u✳ ❑❡r ❥❡

Tf ∈ B(X)✱ ❥❡ ATf − Tf A ∈ B(X) ③❛ ✈s❛❦ f ∈ X∗✳ ❖↔✐t♥♦ ❥❡ (ATf −

Tf A)x = f (x)Au − f(Ax)u ③❛ ✈s❛❦ x ∈ X✳ P♦t❡♠ s❧❡❞✐

|f(Ax)| ≤ ATf − Tf A x + f Au x ,

❦❛r ♣♦♠❡♥✐✱ ❞❛ ❥❡ f ◦ A ∈ X∗ ③❛ ✈s❛❦ f ∈ X∗✳ ●❧❡❞❡ ♥❛ ♥❛❧♦❣♦ ✹✼ s❧❡❞✐

➸❡❧❡♥♦✳

✺✶✳ ◆❛❥ ❜♦❞♦ X✱ Y ✐♥ Z ❇❛♥❛❝❤♦✈✐ ♣r♦st♦r✐✱ A : Y → Z ❧✐♥❡❛r❡♥ ♦♣❡r❛t♦r

✐♥ T ∈ B(X, Y ) s✉r❥❡❦t✐✈❡♥ ♦♣❡r❛t♦r✳ Pr❡❞♣♦st❛✈✐♠♦✱ ❞❛ ❥❡ AT ③✈❡③❡♥

♦♣❡r❛t♦r✳ P♦❦❛➸✐t❡✱ ❞❛ ❥❡ ♣♦t❡♠ t✉❞✐ A ③✈❡③❡♥ ♦♣❡r❛t♦r✳

◆❛♠✐❣✳ ◆❛❥ ❜♦ {yn}n∈N ⊂ Y ③❛♣♦r❡❞❥❡ ③ ❧✐♠✐t♦ 0 ✐♥ {Ayn}n∈N ⊂ Z

③❛♣♦r❡❞❥❡ ③ ❧✐♠✐t♦ z✳ P♦ ✐③r❡❦✉ ♦ ③❛♣rt❡♠ ❣r❛❢✉ ③❛❞♦➨↔❛ ♣♦❦❛③❛t✐✱ ❞❛

❥❡ z = 0✳ ❑❡r ❥❡ T s✉r❥❡❦t✐✈❡♥ ♦♣❡r❛t♦r✱ ♦❜st❛❥❛ {xn}n∈N ⊂ X✱ ❞❛ ❥❡

T xn = yn✳ ●❧❡❞❡ ♥❛ ♥❛❧♦❣♦ ✸✾ ❥❡ {xn}n∈N ❦♦♥✈❡r❣❡♥t♥♦ ③❛♣♦r❡❞❥❡ ③ ❧✐♠✐t♦

0✳ ■③ t❡❣❛ s❧❡❞✐ 0 = AT limn→∞ xn = limn→∞ AT xn = limn→∞ Ayn = z✳

✺✷✳ ◆❛❥ ❜♦❞♦ X1✱ X2 ✐♥ X3 t❛❦✐ ③❛♣rt✐ ♣♦❞♣r♦st♦r✐ ❇❛♥❛❝❤♦✈❡❣❛ ♣r♦st♦r❛ X✱

❞❛ ❥❡ X = X1 ⊕ X2 ⊕ X3✳ Pr❡❞♣♦st❛✈✐♠♦✱ ❞❛ s♦ Pi : X → Xi✱ i = 1, 2, 3✱

✹✾

t❛❦✐ ♣r♦❥❡❦t♦r❥✐✱ ❞❛ ❥❡ ImPi = Xi t❡r Pi|Xj = 0✱ i = j✳ P♦❦❛➸✐t❡✱ ❞❛ s♦

P1, P2, P3 ♦♠❡❥❡♥✐ ♦♣❡r❛t♦r❥✐✳

◆❛♠✐❣✳ P♦❦❛➸✐♠♦✱ ❞❛ ❥❡ P1 ③❛♣rt ♦♣❡r❛t♦r✳ ◆❛❥ ❜♦ {xn}n∈N ⊂ X

③❛♣♦r❡❞❥❡ ③ ❧✐♠✐t♦ 0 ✐♥ {P1xn}n∈N ⊂ X ③❛♣♦r❡❞❥❡ ③ ❧✐♠✐t♦ y✳ P♦ ✐③r❡❦✉

♦ ③❛♣rt❡♠ ❣r❛❢✉ ③❛❞♦➨↔❛ ♣♦❦❛③❛t✐✱ ❞❛ ❥❡ y = 0✳ ●❧❡❞❡ ♥❛ ♣r❡❞♣♦st❛✈❦♦

❧❛❤❦♦ ✈s❛❦ x ∈ X ③❛♣✐➨❡♠♦ ❦♦t x = x1 + x2 + x3✱ x1 ∈ X1✱ x2 ∈ X2✱

x3 ∈ X3✳ P♦t❡♠ ❥❡ P1x = P1x1 + P1x2 + P1x3 = P1x1 = x1. P♦❞♦❜♥♦

✈✐❞✐♠♦✱ ❞❛ ❥❡ P2x = x2 ✐♥ P3x = x3✳ ❚♦r❡❥ ❥❡ x = P1x + P2x + P3x✳

❯♣♦➨t❡✈❛❥♠♦ ♣r❛✈❦❛r ♣♦❦❛③❛♥♦✿ ✐③ ❡♥❛❦♦st✐ xn = P1xn + P2xn + P3xn

s❧❡❞✐ 0 = y + z✱ ❦❥❡r ❥❡ y ∈ X1 ✐♥ z = limn→∞(P2xn + P3xn)✳ ❚♦r❡❥ ❥❡

0 = P1y + P1z = P1y = y. P♦❞♦❜♥♦ ♣♦❦❛➸❡♠♦✱ ❞❛ st❛ P2 ✐♥ P3 ③❛♣rt❛

♦♣❡r❛t♦r❥❛✳

✺✸✳ ◆❛❥ ❜♦ A ❦♦♠♣❧❡❦s♥❛ ❇❛♥❛❝❤♦✈❛ ❛❧❣❡❜r❛ ③ ✐♥✈♦❧✉❝✐❥♦ ∗✳ P♦❦❛➸✐t❡✱ ❞❛

❥❡ ∗ ③✈❡③♥❛ ♣r❡s❧✐❦❛✈❛ ♥❛t❛♥❦♦ t❡❞❛❥✱ ❦♦ ❥❡ H(A) = {x ∈ A | x∗ = x}

③❛♣rt❛ ♠♥♦➸✐❝❛✳

❘❡➨✐t❡✈✳ ✭=⇒) Pr❡❞♣♦st❛✈✐♠♦✱ ❞❛ ❥❡ ✐♥✈♦❧✉❝✐❥❛ ∗ ③✈❡③♥❛ ♣r❡s❧✐❦❛✈❛✳ ◆❛❥

❜♦ {hn}n∈N ❦♦♥✈❡r❣❡♥t♥♦ ③❛♣♦r❡❞❥❡ ✈ H(A) ③ ❧✐♠✐t♦ h✳ P♦t❡♠ ❥❡

h = lim hn = lim h∗ = ( lim h

n

n)∗ = h∗.

n→∞

n→∞

n→∞

❙ t❡♠ s♠♦ ♣♦❦❛③❛❧✐✱ ❞❛ ❥❡ h ∈ H(A)✱ ❦❛r ♣♦♠❡♥✐✱ ❞❛ ❥❡ H(A) ③❛♣rt❛

♠♥♦➸✐❝❛✳

✭⇐=✮ Pr❡❞♣♦st❛✈✐♠♦ s❡❞❛❥✱ ❞❛ ❥❡ H(A) ③❛♣rt❛ ♠♥♦➸✐❝❛✳ ◆❛❥ ❜♦ {xn}n∈N ⊂

A ③❛♣♦r❡❞❥❡ ③ ❧✐♠✐t♦ x ✐♥ {x∗n}n∈N ⊂ A ③❛♣♦r❡❞❥❡ ③ ❧✐♠✐t♦ y✳ ●❧❡❞❡

♥❛ ✐③r❡❦ ♦ ③❛♣rt❡♠ ❣r❛❢✉ ③❛❞♦➨↔❛ ♣♦❦❛③❛t✐✱ ❞❛ ❥❡ y = x∗✳ ❖↔✐t♥♦ ❥❡

limn→∞(xn + x∗ ) = x + y✳ ❑❡r ❥❡ H(A) ③❛♣rt❛ ♠♥♦➸✐❝❛ ✐♥ ❥❡

n

{xn +

x∗n}n∈N ⊂ H(A)✱ ❥❡ x + y ∈ H(A)✳ Pr❛✈ t❛❦♦ ✈✐❞✐♠♦✱ ❞❛ ❥❡ {i(x∗n −

xn)}n∈N ⊂ H(A)✳ ❩❛t♦ ❥❡ i(y − x) ∈ H(A)✳ ❚♦r❡❥ ❥❡ x + y = x∗ + y∗

✐♥ i(y − x) = −i(y∗ − x∗)✳ ▼♥♦➸❡♥❥❡ ③❛❞♥❥❡ ❡♥❛❦♦st✐ s ❦♦♠♣❧❡❦s♥✐♠

➨t❡✈✐❧♦♠ i ♥❛s ♣r✐✈❡❞❡ ❞♦ ✐❞❡♥t✐t❡t❡ x − y = y∗ − x∗✳ ■③ t❡❣❛ s❧❡❞✐✱ ❞❛ ❥❡

y = x∗✳

✺✵

▲✐t❡r❛t✉r❛

❬✶❪ ▼✳ ❍❧❛❞♥✐❦✱ ◆❛❧♦❣❡ ✐♥ ♣r✐♠❡r✐ ✐③ ❢✉♥❦❝✐♦♥❛❧♥❡ ❛♥❛❧✐③❡ ✐♥ t❡♦r✐❥❡ ♠❡r❡✱

❋❛❦✉❧t❡t❛ ③❛ ♥❛r❛✈♦s❧♦✈❥❡ ✐♥ t❡❤♥♦❧♦❣✐❥♦✱ ■♥➨t✐t✉t ③❛ ♠❛t❡♠❛t✐❦♦✱ ✜③✐❦♦ ✐♥

♠❡❤❛♥✐❦♦✱ ▲❥✉❜❧❥❛♥❛ ✶✾✽✺

❬✷❪ ❙✳ ❑✉r❡♣❛✱ ❋✉♥❦❝✐♦♥❛❧♥❛ ❛♥❛❧✐③❛✱ ❊❧❡♠❡♥t✐ t❡♦r✐❥❡ ♦♣❡r❛t♦r❛✱ ➆❦♦❧s❦❛ ❦♥✲

❥✐❣❛✱ ❩❛❣r❡❜ ✶✾✽✶

❬✸❪ ■✳ ❱✐❞❛✈✱ ▲✐♥❡❛r♥✐ ♦♣❡r❛t♦r❥✐ ✈ ❇❛♥❛❝❤♦✈✐❤ ♣r♦st♦r✐❤✱ ❯♥✐✈❡r③❛ ✈ ▲❥✉❜❧❥❛♥✐✱

■♥➨t✐t✉t ③❛ ♠❛t❡♠❛t✐❦♦ ✜③✐❦♦ ✐♥ ♠❡❤❛♥✐❦♦✱ ❋❛❦✉❧t❡t❛ ③❛ ♥❛r❛✈♦s❧♦✈❥❡ ✐♥

t❡❤♥♦❧♦❣✐❥♦✱ ▲❥✉❜❧❥❛♥❛ ✶✾✽✷

✺✶