# 0x023 Functional Analysis

## 1. Banach Space

### Normed Vector Space

Definition (norm) A norm on a vector space $$V$$ is a function $$|| \cdot ||: V \to [0, \infty)$$ such that

• (positive definite) $$||f|| = 0 \iff f=0$$
• (homogeneity) $$||\alpha f|| = |\alpha| ||f||$$
• (triangle inequality) $$||f+g|| \leq ||f||+||g||$$

not norm

Let $$\mathcal{L}^1(R)$$ be the vector space of Borel measurable function $$f: R \to F$$ such that $$\int |f| d\lambda < \infty$$ where $$\lambda$$ is the Lebesgue measure. define $$|| \cdot||_1$$ on $$\mathcal{L}^1(R)$$ by

$||f||_1 = \int |f| d\lambda$

It is not a norm as it does not satisfy the positive definte property. Consider an nonempty Borel subset $$E$$ of $$R$$ with Lebesgue measure 0, then $$|\chi_E| = 0$$ but $$\chi_E \neq 0$$. This problem can be solved by constructing $$L^p$$ space.

Definition (Banach space) A complete normed vector space is called a Banach space.

Theorem (Baire) A complete metric space is not the countable union of closed subsets with empty interior

Recall the linear map is defined as follows:

Definition (linear map) Suppose $$V,W$$ are vector spaces, a function $$T: V \to W$$ is called linear if

• $$T(f+g) = Tf + Tg$$
• $$T(\alpha f) = \alpha T(f)$$

In analysis, we usually focus on a subset of bounded linear functions

Definition (bounded linear map) Suppose $$V,W$$ are normed vector spaces and $$T: V \to W$$ is a linear map, the norm of $$T$$, denoted $$||T||$$, is defined by

$||T|| = \sup \{ ||Tf|| : f \in V , ||f|| \leq 1 \}$

$$T$$ is called bounded if $$||T|| < \infty$$

The set of bounded linear maps from $$V$$ to $$W$$ is denoted $$\mathcal{B}(V,W)$$

The wikipedia's definition is a bit easier to understand: bounded linear operator is a linear transformation between topological vector spaces $$X \to Y$$ where $$X,Y$$ are normed vector spaces, then $$L$$ is bounded if and only if there exists some $$M>0$$ such that for all $$x\ in X$$

$\|Lx\|_{Y}\leq M\|x\|_{X}$

The smallest such $$M$$ is called the operator norm of $$L$$ and denoted by $$\|L\|$$

Theorem (continuity is equivalent to boundedness) A linear map from one normed vector space to another is continuous iff bounded.

### Linear Functional

Definition (linear functional) A linear functional on a vector space $$V$$ is a linear map from $$V \to R$$

## 2. LP Space

Definition ($$||f||_p$$) Suppose that $$(X, S, \mu)$$ is a measure space, $$0 < p < \infty$$ and $$f: X \to F$$ is $$S$$-measurable. Then the p-norm of $$f$$ is defined by

$||f||_{p} = \Big( \int |f|^p d\mu \Big)^{1/p}$

Note that the exponent is to make sure $$||af||_p = |a| || f||_p$$

Definition (essential supremum) The essential supremum of $$f$$ is

$||f||_{\infty} = \inf \{ t > 0 | \mu(\{ x \in X | |f(x)| > t \} ) = 0 \}$

counting measure

Suppose $$\mu$$ is counting measure on $$Z^+$$, if $$a=(a_1, a_2, ...)$$ is a sequence in $$F$$ and $$0 < p < \infty$$, then

$\| a \|_p = (\sum^{\infty}_{k=1} |a_k|^p)^{1/p}$

and

$\| a \|_{\infty} = \sup \{ a_k | k \in Z^+ \}$

Definition (Lebesgue space, $$\mathcal{L}^p(\mu)$$) Suppose $$(X, S, \mu)$$ is a measure space. The Lebesgue space $$\mathcal{L}^p(\mu)$$ is denoted by the set of S-measurable function $$f$$ such that

$\| f \|_p < \infty$