ACIT4330-Page/content/Lectures/Lecture 5.md

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5	2025-01-20

[!info] A normed space is a Banach Space when the corresponding metric space is complete. Any normed space can be completed to a Banach Space, see the real numbers from \mathbb{Q}.

[!example] Example: \mathbb{C}^n vector space, then

1. \| \, v \, \|_{1} \equiv \sum_{k=1}^{n} |v_{k}|, \; v=(v_{1}, \dots, v_{n})
2. \| \, v \, \|_{\infty} \equiv \text{sub}_{k=1,\dots,n}\{ |v_{k} | \}

Which are norms on \mathbb{C}^n

Example

[!question] Consider C_{c}(X) \subset \Pi_{X}\mathbb{C} = \{ f : X \to C \} on functions X -> \mathbb{C} that are non-zero for finitely only many x \in X.

Then C_{c}(X) is a vector space under op.

f \neq \text{only on} \, Y \, \subset \, X q \neq \, \text{only on} \, Z \subset X \implies f+q \neq 0 \, \text{only on} \, Y \cup Z a \times f \neq \, \text{only on} \, Y

with \delta_{x}(y) = \begin{cases}1, & x=y\\ 0, & x\neq y\end{cases} Has linear basis \{ \delta_{x} \}, x \in X

Proof

Given that f \in C_{c}(X) then

f = \sum_{x \in X} f(x) \times \delta_{x} is a finite sum, and the only possibility.

QED

So \text{dim} C_{c} (X) = |X|.

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Definite norms on C_{c}(X) by \| \, f \, \|_{1} = \sum_{x \in X} | f(x) | and \| \, f \, \|_{\infty} = \sup_{x \in X} | f(x) | when |X| = n we recover \mathbb{C}^n.

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We write $V \simeq W$and say that V and W are Isomorphic

1. As sets if \exists Bijective f : V \to W
2. As vector spaces if in addition f is linear; f(a \times u + b \times v) = a \times f(u) + b \times f(v), \; \forall a,b \in \mathbb{C}, \; u,v \in V
3. As Normed vector Space if in addition f is Isometric; \| \, f(v) \, \| \, = \| \, v \, \|, \; \forall v \in V. f should preserve all relevant structure.

Hilbert Spaces

Triangle Inequality

Follows from the Cauchy-Schwarz Inequality

[!example] Example on C_{c}(X) then

(f \mid g) = \sum_{x \in X} f(x) \times \overline{g(x)}

This defines an Inner Product, which can be completed to a Hilbert Spaces.

\| \, f \, \|_{2} = \left( \sum_{x \in X} \mid f(x) \mid^2 \right)^{\frac{1}{2}}

[!note] The Inner Product here

\mathbb{C}^{n} (u \mid v) = \sum_{k = 1}^{n} u_{k} \overline{v_{k}} \mathbb{R}^{n} \| \, u \, \|_{2} = \left( \sum \mid u_{k} \mid^{2} \right)^{\frac{1}{2}} \vec{u} \times \vec{v} = (\vec{u} \mid \vec{v})