1. A (i) 令 X 为向量空间。说 k k 是一个范数是什么意思

(ii) 对于 1 p 1，准确地说出序列是什么意思
x = (x1; x2; : : : ) 的复数属于`p
.写下定义
`p 上的（标准）范数
. [3 分]
B 让​​ 1 < p; r < 1. 给出一个充分必要条件（在
p 和 r) 的项，用于连续嵌入 `p
`r

C 如果对于每个 > 0 存在 > 0，则称 Banach 空间为一致凸空间

2 (x + y)k > 1 意味着 kx yk < “。

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

[11分]
2. A 让 1 p < 1. 准确地说明函数
f : [0; 1]！ C属于Lp
([0; 1])。定义这个空间的（通常）范数。
[3 分]
B 是否存在函数 f 2 L1
(R) 使得 limsupx!1 f(x) = 1？

C 举一个集合 A L2 的例子
([0; 1])，在 L2 中是无界的
([0; 1]),

([0; 1])。 [11分]

3. A (i) 陈述复线性空间上内积的公理。
(ii) 陈述（但不证明）希尔伯特中的 Riesz 表示定理

B 令 M 是希尔伯特空间 H 的一个闭子空间，并假设 T : M ！
C 是有界线性泛函。证明存在一个有界线性

C 令 X 为希尔伯特空间。证明下列断言为真或

(i) 如果 M 是 X 的稠密线性子空间，那么 M？ = f0g。
(ii) 如果 M 是 X 的线性子空间，使得 M? = f0g，那么 M 是稠密的。

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]

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