Functional Analysis

§1 Metric Space

§1.1 Contraction Mapping Principle

Definition:(Metric space) Let be a non-empty set. is called metric space if there is a function on into s.t.

• ;
• ;
• . ().

Here is call metric on . A metric space with as metric is denoted as .

$${\sqrt{ }}$$

§1.2 Completion

Definition: A sequence is called a fundamental sequence iff s.t. .

Definition: A sequence in converges to iff ().
We denote it as .

Definition: A subset of is closed iff .

Definition: A subset of is called sequential-compact(abbr.: s.c.) iff and is called self-sequential-compact iff .

Definition: A subset is complete iff is a fundamental sequence .

Theorem: On , boundedness compactness ( completely bounded boundedness).

§1.3 Sequential-Compactness

Theorem:

Definition: is a -net iff . If is finite, then is called a finite -net.

Remark: .

Definition: is called completely bounded(abbr.: c.b.) iff , net of .

Theorem: (Hausdorff) is a subset of a metric space .

• is sequentially compact is completely bounded.
• is completely bounded and complete is sequentially compact.

Proof:
“ is sequentially compact is completely bounded”:
Suppose otherwise, then s.t. doesn’t have -net. Then . Since is non-empty, we have , so s.t. . And for , we have s.t. . recursively we get a sequence s.t. . So which is contradictory to the assumption(if s.t. , then for sufficiently large and , we have ). So is completely bounded.
“ is completely bounded and complete is sequential compact.”(completeness Cauchy sequence -nets):
Let be a sequence in (finite or infinite). For , there exists a -net , then there must be a subsequence of and s.t. (principle of pigeon hole). Thus we gain a list:

$$\begin{matrix} \left( m_{0}, \frac{1}{1} \right): & x^{1}_{1} & x^{1}_{2} & \ldots & \\ \left( m_{1}, \frac{1}{2} \right): & x^{2}_{1} & x^{2}_{2} & \ldots & \\ \left( m_{3}, \frac{1}{3} \right): & x^{3}_{1} & x^{3}_{2} & \ldots & \\ \vdots & \vdots & \vdots \end{matrix}$$

Let , then for , . So is a Cauchy sequence which is convergent since is complete.

Remark: s.c. is stronger than c.b.(列紧强于完全有界)

Definition: A metric space having a dense subset is called separable. (可分)

Theorem: Completely bounded metric space is separable.

Proof: For , pick -net . Then is a dense subset of the metric space.

Definition:(Compactness)A topology space is called compact iff family of open sets s.t. , finite subset of s.t. .

Remark: A compact space can be treated as a finite set in an aspect.

Theorem: is a metric space. is a subset of . is compact iff is self sequential compact.

Proof:
: Firstly, we prove that is closed. For . . Since is compact, there exists s.t. . Secondly, we prove that is s.c.. Suppose the opposite i.e. s.t. for all subsequence of , it doesn’t converges. We claim that is a closed set. This is because for all subset of s.t. . Let . Then is open. Thus . (Remark: : remove all points but ) Since is compact, there is a finite set of , s.t. which is impossible since which is contradictory to that . Combine these results. We gain that is self-s.c..
: Otherwise, suppose is a family of open sets that covers . There is no finite sub-family of s.t. covers . Since is s.c., is c.b.. For all , there is a -net . So there must be a s.t. can not be covered by finite sets of . Collect these . Since is self-s.c., subsequence {. Suppose for some . Then there exists s.t. . Let be great enough(e.g. and makes ) so that and which is a contradiction.

Theorem: (Arzela-Ascoli): is a sequential-compact set iff is uniformly bounded(bounded w.r.t the norm ) and isometric continuous).

Example: 设是有界开凸集. 若, 是两个给定的正数, 则集合${F \triangleq \{\varphi \in C^{(1)}(\bar{\Omega})\vert\ \lVert \varphi(x) \rVert \leq M_{1}, \lVert \operatorname{grad}\varphi(x) \rVert \leq M_{2}\,\forall\,x \in \Omega \}}$是上的一个列紧集, 其中表示上的连续可微函数全体.

Proof: 由Arzela-Ascoli定理只需说明一致有界和等度连续. 等度连续由保证.

§1.4 Space

Definition: , is filed of complex numbers(field of real numbers). If:

• is an Abel group w.r.t .
• There is a scalar multiplication on s.t.
  • .
  • ;
  • , .

Remark: A communicative module under .

Definition: (generated sub linear space) is a sub linear space of . Let be a subset of s.t. If for any sub linear space containing , . We say that is generated by .

Remark: exists uniquely. since firstly, is unique and secondly is a generated sub linear space by . .

Definition: (Add linearity to metric quasi norm) Quasi norm on linear space is a function on to s.t.

• ;
• ;(triangle inequality)
• ;(symmetry)
• as or for all and .(continuity)

Definition: ( space) A linear space with a quasi norm on it is called space. Denote as as .

Definition: Complete space is call space. (Here completeness means: as as .)

Example: ( is a compact metric space) is a space with quasi norm .

Example: is a space with quasi norm .

Example: . Here is a metric space. is a space with quasi norm .

as .
.
as for any .
Let be a separable space, . Then is bounded has a weakly convergent subsequence.

Example: 表示上一切连续函数全体, 并令${\lVert u \rVert = \sum_{k=1}^{\infty} \frac{1}{2^{k}} \frac{\max_{\lvert x \rvert \leq k}\lvert u(x) \rvert}{1 + \max_{\lvert x \rvert \leq k} \lvert u(x) \rvert}}.$其中. 是一个准范数, 是一个空间.

1.5 Norm

Definition:(Norm)(homogeneity + quasi-norm) A norm on linear is a map on to s.t.

• ;
• ;
• .(essential)

Definition: ( space) A normed linear space which is complete is called a space. (Completion: .

Example(Classic): space is complete when . When is -finite, is complete.

Proof: For , …
For , …

Remark:
Claim:
as .

Proof:
: means , s.t. , . i.e. . So, . Since sequence is monotone, it converges iff a subsequence of it converges. For any we can find a s.t. . Thus as .
: For all , as . , . , , , . (denote as ) . Let , then . Then , , , . Then . . .

Claim: If , as subsequence of , denoted as , subsequence of , denoted as s.t. as .

Proof:
as .
:
, as . Choose s.t. . And for all , . So we can get a sequence s.t. and . , s.t. . So . as . By the former claim, .
: Suppose the opposite, there exists and and a sequence s.t. . Rewrite as . Then for any subsequence of , , which is contradictory to the assumption. (Notice that all subsequence of as )

*

Theorem: (Banach-Steinhaus) Let be a Banach space and be a space, be a dense subset of . be a sequence in . Then $\exists f \text{ s.t. }f_{n}(x) \to f(x), \forall x \in X$ iff

• ;
• .

Theorem: Let be a Banach space, , iff

• ;
• For a dense subset in , .

$$\begin{CD} A @>a>> B\\ @VVbV @VVcV\\ C @>d>> D \end{CD}$$