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
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
\(T\) 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 \(x\ in X\)
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
Note that the exponent is to make sure \(||af||_p = |a| || f||_p\)
Definition (essential supremum) The essential supremum of \(f\) is
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
and
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
3. Hilbert Space
Inner Product Space
Orthogonality
Orthonormal bases
Linear Map on Hilbert Spaces
Adjoints and Invertibility
Spectrum
Compact Operator
Spectral Theorem
4. Reference
[1] Axler, Sheldon. "Measure, Integration & Real Analysis." (2020): 411.