These are my notes right now for Section 5 of Chapter 5, Hilbert Spaces. (Dated April 12, 2022)

First we introduce the definition of an inner product. First let \(\mathcal{H}\) be a complex vector space. An \(\textbf{inner product}\) on \(\mathcal{H}\) is a map \((x,y) \rightarrow \braket{x,y}\) from \(\mathcal{X} \times \mathcal{X} \rightarrow \mathbb{C} \) such that : (i): \(\braket{ax+by,z} = a \braket{x,z} +b \braket{y,z} \) for all \(x,y,z \in \mathcal{H}\) and \(a,b \in \mathbb{C}\). (ii): \(\braket{y,x} = \overline{\braket{x,y}}\) for all \(x,y \subset \mathcal{H}\) (iii): \(\braket{x,x} \in (0,\infty)\) for all nonzero \(x\in \mathcal{X}\)

A complex vector space with an inner product is called a \(\bf{\textrm{pre-Hilbert space}}\). If \(\mathcal{H}\) is a pre-Hilbert Space, for \(x \in \mathcal{H}\), we define \(\begin{gather*} { \|x \| = \sqrt{\braket{x,x}}} \end{gather*}\)

\(\bf{\textrm{The Schwarz Inequality}} : \| \braket{x,y} \| \leq \|x\| \|y\|\) for all \(x,y \in \mathcal{H}\), with equality iff x and y are linearly dependent.

\(\bf{\textrm{Proposition}}:\) The function \(x\rightarrow \|x\|\) is a norm on \(\mathcal{H}\)

A pre-Hilbert space that is complete with respect to the norm \(\|x\| = \sqrt{\braket{x,x}}\) is called a \(\textrm{\bf{Hilbert Space}\).

\(\bf{\textrm{Proposition}:}\) If \(x_n \rightarrow x\) and \(y_n \rightarrow y\), then \(\braket{x_n,y_n} \rightarrow \braket{x,y}\)

\(\bf{\textrm{The Parallelorgram Law}}:\) For all \(x,y \in \mathcal{H}\), \(\|x+y\|^2 + \|x-y\|^2 = 2(\|x\|^2 +\|y\|^2)\)

If \(x,y \in \mathcal{X}\), we say that \(x\) is \(\bf{orthogonal}\) to \(y\) and write \(x \perp y\) if \(\|x,y\|=0\). If \(E\subset \mathcal{H} \) we define: \(E^{\perp}={x\in\mathcal{H} : \braket{x,y} = 0, \mathrm{for all } y\in \mathcal{H}}\)

\(\bf{\textrm{The Pythagorean Theorem}:}\) If \(x_1,x_2,..., x_n \in \mathcal{H}\) and \(x_j \perp x_k \) for \(j\neq k\), \(\| \Sigma_{1}^{n}x_j\|^2 = \Sigma_{1}^{n} \|x_j\|^2\)

\(\bf{\textrm{Theorem 5.25}:}\) If \(f \in \mathcal{H^*}\) there is a unique \(y\in \mathcal{H}\) such that \(f(x)=\braket{x,y}\) for all \(x \in X\)

A subset \(\{u_\alpha\}_{\alpha \in A}\) of \(\mathcal{H}\) is called \(\mathbf{orthonormal}\) if \(\|u_{\alpha} \| -1 \) for all \(\alpha\) and \(u_\alpha \perp u_\beta\) whenever \(\alpha \neq \beta\).

\(\bf{\textrm{Gram-Schmidt process}:}\) This is a process for converting a linearly independent sequence in a Hilbert space \(\mathcal{H}\)

\(\bf{\textrm{Bessel's Inqeuality}}\)

\(\bf{\textrm{Theorem 5.27}}\)

\(\bf{\textrm{Orthonormal Basis}}\)

\(\bf{\textrm{Proposition 5.28}}\)

\(\bf{\textrm{Proposition 5.29}}\)

\(\bf{\textrm{Unitary Map}}\)

\(\bf{\textrm{Proposition 5.30}}\)

Section 1 Chapter 6 (Dated April 14, 2022)

First we introduce the definition of the \(L^p\) norm. Given a fixed measure space \((X,\mathcal{M},\mu)\). If \(f\) is a measureable function on \(X\) and \(0<p<\infty\), we define:

\(\|f\|_p = [ \int |f|^p d\mu ]^{1/p}\)

with the added possibility that \(\|f\|_p = \infty\). We then define:

\(L^{p}(X, \mathal{M}, \mu) = {f: X \rightarrow \mathbb{C} : f \mathrm{is measurable and} \|f\|_p < \infty}\)

This is abbreviated by \(L^p(\mu), L^p(X), L^p\) whenever the domain is understood.

There are some interesting properties for the norm depending on the value of p. For example, the triangle inequality does not hold when \(0<p<1\).

Proof: Consider \(a>0,b>0\) and \(0<p<1\).

(6.1-6.8)

Next time we will start on (6.9)

Dated (April 19, 2022 We've discussed 6.9 - 6.20

Dated (April 21, 2022) We reviewed the things we learned on Tuesday.

MM: Come back later and fill this in.