Table of Contents
1. Support
1.1. Set-Theoretic Support
- Support of a real-valued function \(f\) with domain \(X\) is: \[ \operatorname{supp}(f) = \{x\in X : f(x) \neq 0\}. \]
- The smallest subset of \(X\) such that \(f\) is zero on the subset's complement.
1.2. Closed Support
- Support
- The (closed) support of \(f\) on a ../geometry/Topology.html#org98e01d0 \(X\) is: \[ \operatorname{supp}(f) = \operatorname{cl}_X(\{x\in X : f(x) \neq 0\}). \]
1.3. Compact Support
- Compact closed support
1.4. Essential Support
- For a topological space \(X\) with a \(\mu\), the essential support \(\operatorname{ess\ supp}(f)\) of a \(f:X\to \mathbb{R}\) is the smallest subset \(F\) of \(X\) such that \(f\) is zero \(\mu\)-almost everywhere outside \(F\).
1.4.1. Examples
- For the ../Glossary.html#orge67f855 \(f: [0, 1] \to \mathbb{R}\),
and \([0, 1]\) equipped with the :
- The support of \(f\) is the entire interval \([0, 1]\),
- but the essential support of \(f\) is empty.
2. Bounded Operator
2.1. Bounded Linear Operator
The space of bounded linear operator between two topological vector spaces \( X, Y \) is denoted by \( \mathcal{B}(X,Y) \).
- It is normed vector space.
3. Continuous Map
A function \( f\colon X\to Y \) between topological spaces \( X, Y \) is called continuous if \[ A \subseteq Y \text{ open in $Y$} \implies f^{-1}[A] \subseteq X \text{ open in $X$}. \]
4. Open Map
A function \( f\colon X\to Y \) from metric spaces \( X,Y \) is called open if \[ A \subseteq X \text{ open in $X$} \implies f[A] \subseteq Y \text{ open in $Y$}. \]
5. Open Mapping Theorem (Banach-Schauder Theorem)
For a bounded linear operator \( T \in \mathcal{B}(X,Y) \) between Banach spaces \( X, Y \) \[ T\text{ surjective} \implies T \text{ open map}. \]
6. Spectrum
For the most part is does not work?
7. Hahn-Banach Theorem
There exists an extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space.