本次英国代写主要为infinite dimensional vector space限时测试
7CCMMS05T公制和Banach空间（MSc）
7CCMMS05U公制和Banach空间（MSci）
2020年夏季
允许时间：两个小时
所有问题都带有相等的分数。
对于四个问题的完整答案，将获得满分。
如果尝试了四个以上的问题，则只有最佳的四个
数数。
您可以查阅讲义。
1. A（i）令X为向量空间。说k k是范数是什么意思
在X上？
（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
保持真实。证明
您的条件既必要又充分。直接从
定义。 [11分]
如果每个“> 0都存在> 0，则一个Banach空间称为统一凸
这样kxk = kyk = 1和k1
2（x + y）k> 1表示kx yk <“。
证明任何希尔伯特空间都是一致凸的。
您可以在没有证明的情况下使用平行四边形法则
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？
为您的回答辩护。 [11分]
C给出一个集合A L2的例子
（[0; 1]），在L2中不受限制
（[0; 1]），
但限于L1
（[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
空间。 [3分]
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
功能S：H！ C，使得M上的S = T，kSk = kTk。
您可以使用课程中有关希尔伯特空间的任何结果，但您可以
不要使用Hahn-Banach定理。 [8分]
C令X为希尔伯特空间。证明以下断言是正确的或
提供表明它们为假的反例：
（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：
证明定义良好的有界线性图并计算其范数。
您可以使用H old的不等式，但课程中不会有其他结果。
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 sucient 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 sucient. 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?
Justify brie y your answer. [11 marks]
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 Holder’s inequality but no other results from the course.