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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

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


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

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

\[\| f \|_p < \infty\]

3. Hilbert Space

Inner Product Space


Orthonormal bases

Linear Map on Hilbert Spaces

Adjoints and Invertibility


Compact Operator

Spectral Theorem

4. Reference

[1] Axler, Sheldon. "Measure, Integration & Real Analysis." (2020): 411.