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0x023 Functional Analysis

1. Banach Space

1.1. 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.

Banach Space

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

Banach spaces

the vector space \(C([0,1])\) (all continuous function in \([0,1]\)) with the norm \(\| f \| = \sup_{[0,1]} |f|\) is a Banach space

the vector space \(\mathcal{l}\)^1 with norm defined by \(\| (a_1, a_2, ...) \|_1 = \sum_{k=1}^{\infty} |a_k|\) is 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.

1.2. 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

3.1. Inner Product Space

3.2. Orthogonality

3.3. Orthonormal bases

4. Linear Map on Hilbert Spaces

4.1. Adjoints and Invertibility

4.2. Spectrum

4.3. Compact Operator

4.4. Spectral Theorem

5. Reference

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