基本信息

---

下载

文档摘要

---

★

2024-2025

1.

2.xzqbear

3.100

4..

5.20241231

1.15

Letbeanynon-emptysetandbeaBanachspaceover?.Let(,)bethevectorspaceof

boundedcontinuousfunctionsfromtowiththenorm

‖‖=sup‖()‖

∈

.Showthat(,)isaBanachspace.

2.15

×T

Let(?)bethespaceofall?matrix.Define?,?=tr().Prove??,??isaninnerproduct

on(?).

3.15

Let1?∞andconsider∶?→[0,∞)definedby

∞

()=∑,?∈()∈?.

[?1,)∈?

=1

Provethat‖‖=‖‖forany∈?.

4.15

22

Let∶[0,1]→?beacontinuousfunctionand∶[,]→[0,1]definedby

1

()=()∫()()d

0

Provethatisself-adjointandpositive.

5.15

Considerasa-finitemeasurespace,and∈∞().Defineasalinearoperator:

()=()()

2?

∈()and∈arearbitrary.Findadjointoperator.

1??

6.15