Hilbert Spaces and Related Concepts

Inner product and normed spaces

Definition of inner product

Let $X$ be a vector space over $\mathbb{R}$, An inner product on $X$ is a function $\langle\cdot,\cdot\rangle: X \times X\to\mathbb{R}$ such tat for all $x,y, z \in X$ and $\alpha, \beta \in\mathbb{R}$,

1. $\langle x,x\rangle\geq 0$ and $\langle x,x\rangle = 0$ if and only if $x=0,$
2. $\langle\alpha x + \beta y, z\rangle = \alpha\langle x,z\rangle + \beta\langle y, z\rangle,$
3. $\langle x,y\rangle = \langle y,x\rangle.$

A vector space $X$ with inner product is called an inner product spaces.

Example

Let $\Omega \sub\mathbb{R}^n.$ If $f,g\in L^2(\Omega),$ then $fg\in L^1(\Omega)$ and the function $\langle\cdot,\cdot\rangle: L^2(\Omega) \times L^2(\Omega)\to\mathbb{R}$ defined by

$$\langle f,g\rangle=\int_\Omega fg d\mu,$$

is an inner product on $L^2(\Omega).$

Something extremely important is that you can induce a norm by an inner product, in fact, let $X$ be an inner product space and $x, y\in X$, then, the function $\lVert \cdot \rVert: X \to \mathbb{R}$ defined by $\lVert x \rVert:\langle x,x\rangle^{\frac{1}{2}}$, is a norm on $X$, and furthermore, X is a normed space.

Hilbert Spaces

Definition of Hilbert Space

An inner product space in which every Cauchy sequence (with respect to the norm induced by the inner product) converges to an element of the space is called a Hilbert space. This means the space is complete.

Example

As the $L^p(\Omega)$ is a complete space then the previous example $L^2(\Omega)$ is a Hilbert space.

Riesz Theorem

If $H$ is a Hilbert space and $T\in H'$ then there us a unique $y\in H$ such that $T(x)=\langle x,y\rangle$ for all $x\in H$. Moreover $\lVert T \rVert=\lVert y \rVert.$

Example

Consider the Hilbert space $H = \mathbb{R}^2$ with the usual inner product $\langle x, y \rangle = x_1 y_1 + x_2 y_2$ and define the linear transformation $T(x_1, x_2) = 3x_1 + 4x_2$.

According to the Riesz Representation Theorem, there must exist a unique vector $y = (y_1, y_2) \in \mathbb{R}^2$ such that $T(x_1, x_2) = \langle (x_1, x_2), (y_1, y_2) \rangle = x_1 y_1 + x_2 y_2$

Therefore, the representing vector is $y = (3,4)$ and the norm of the transformation is $\|T\| = \|y\| = \sqrt{3^2 + 4^2} = 5$

The transformation $T(x_1, x_2) = 3x_1 + 4x_2$ can be written as $T(x) = \langle (x_1,x_2), (3,4) \rangle$

Example

Let $H = L^2([0,1])$ and define the linear transformation $T(f) = \int_0^1 f(x)\, x\ d\mu.$

By the Riesz Representation Theorem, there exists a unique function $g \in L^2([0,1])$ such that

$$T(f) = \langle f, g \rangle = \int_0^1 f(x) g(x)d\mu$$

Comparing the integrands, we deduce that $g(x) = x.$

The norm of the transformation is:

$$\|T\| = \|g\| = \left( \int_0^1 x^2 dx \right)^{1/2} = \sqrt{\tfrac{1}{3}}$$

The transformation $T(f) = \int_0^1 f(x)x\, dx$ can be represented as $T(f) = \langle f, x \rangle$

Definition of Weak Convergence

Let $H$ be a Hilbert space and let $\{x_n\} \subset H$ be a sequence. We say that $\{x_n\}$ converges weakly to $x \in H$, and we write $x_n \rightharpoonup x,$ if and only if,

$$\langle x_n, y \rangle \to \langle x, y \rangle \quad \text{for all } y \in H.$$

This definition relies on the Riesz Representation Theorem, which ensures that every bounded linear transformation on $H$ can be expressed as an inner product with a fixed element of $H$.