7CCMMS05T公制和Banach空间（MSc）
7CCMMS05U公制和Banach空间（MSci）
2020年夏季

1. A（i）令X为向量空间。说k k是范数是什么意思

（ii）对于1 p 1，请准确说明说该序列是什么意思
x =（x1; x2;：：：）的复数属于`p
。写下定义
p上（标准）规范的
。 [3分]
B令1 <p; r <1.给出必要和充分的条件（在
p和r的项），用于连续嵌入`p
`r

2（x + y）k> 1表示kx yk <“。

kx + yk2
+ kx yk2
= 2kxk2
+ 2kyk2

[11分]
2. A让1 p <1。精确说明说一个函数的含义
f：[0； 1]！ C属于Lp
（[0; 1]）。在此空间上定义（通常）规范。
[3分]
B是否存在函数f 2 L1
（R）使得limsupx！1 f（x）= 1？

C给出一个集合A L2的例子
（[0; 1]），在L2中不受限制
（[0; 1]），

（[0; 1]）。 [11分]

3. A（i）陈述内积在复杂线性空间上的公理。
（ii）在希尔伯特中陈述（但不证明）里斯表示定理

B令M为Hilbert空间H的封闭子空间，并假定T：M！
C是有界线性泛函。证明存在一个有界的线性

C令X为希尔伯特空间。证明以下断言是正确的或

（i）如果M是X的稠密线性子空间，则M？ = f0g。
（ii）如果M是X的线性子空间，使得M？ = f0g，则M为稠密的。
M？表示X中M的正交补码。[14分]
4. A设X为Banach空间。
（i）解释说“ X”是什么意思！ C属于对偶空间
X
。写下2 X范数的定义

（ii）说X是泛指是什么意思？ [3分]
B令1 <p <1并定义：Lp
[1; 1]！ C由
F ！
11
1个
f（x）dx：

7CCMMS05T Metric and Banach Spaces (MSc)
7CCMMS05U Metric and Banach Spaces (MSci)
Summer 2020
Time Allowed: Two Hours
All questions carry equal marks.
Full marks will be awarded for complete answers to FOUR questions.
If more than four questions are attempted, then only the best FOUR will
count.
You may consult lecture notes.

1. A (i) Let X be a vector space. What does it mean to say that k  k is a norm
on X?
(ii) For 1  p  1, state precisely what it means to say that the sequence
x = (x1; x2; : : : ) of complex numbers belongs to `p
. Write down the de nition
of the (standard) norm on `p
. [3 marks]
B Let 1 < p; r < 1. Give a necessary and sucient condition (explicitly in
terms of p and r) for the continuous embedding `p
 `r
to hold true. Prove
that your condition is both necessary and sucient. Argue directly from the
de nitions. [11 marks]
C A Banach space is called uniformly convex if for every ” > 0 there exists  > 0
such that kxk = kyk = 1 and k1
2 (x + y)k > 1  imply that kx yk < “.
Prove that any Hilbert space is uniformly convex.
You may use without proof the parallelogram law
kx + yk2
+ kx yk2
= 2kxk2
+ 2kyk2
:
[11 marks]
2. A Let 1  p < 1. State precisely what it means to say that a function
f : [0; 1] ! C belongs to Lp
([0; 1]). De ne the (usual) norm on this space.
[3 marks]
B Does there exist a function f 2 L1
(R) such that limsupx!1 f(x) = 1?
C Give an example of a set A  L2
([0; 1]), which is unbounded in L2
([0; 1]),
but bounded in L1
([0; 1]). [11 marks]

3. A (i) State the axioms of an inner product on a complex linear space.
(ii) State (but do not prove) the Riesz representation theorem in a Hilbert
space. [3 marks]
B Let M be a closed subspace of a Hilbert space H and suppose that T : M !
C is a bounded linear functional. Show that there exists a bounded linear
functional S : H ! C such that S = T on M and kSk = kTk.
You may use any result about Hilbert spaces from the course but you may
not use the Hahn-Banach theorem. [8 marks]
C Let X be a Hilbert space. Prove that the following assertions are true or
provide counterexamples showing that they are false:
(i) If M is a dense linear subspace of X, then M? = f0g.
(ii) If M is a linear subspace of X such that M? = f0g, then M is dense.
Here M? denotes the orthogonal complement of M in X. [14 marks]
4. A Let X be a Banach space.
(i) Explain what it means to say that  : X ! C belongs to the dual space
X
. Write down the de nition of the norm of  2 X
.
(ii) What does it mean to say that X is re exive? [3 marks]
B Let 1 < p < 1 and de ne  : Lp
[ 1; 1] ! C by
f !
Z 1
1
f(x)dx:
Prove that  is well-de ned bounded linear map and compute its norm.
You may use Holder’s inequality but no other results from the course.

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