15.4 Hilbert spaces

Hilbert spaces

A vector space $S$ with a valid inner product defined on it is called an inner product space , which is also a normed linear space . A Hilbert space is an inner product space that is complete with respect to the norm defined using the innerproduct. Hilbert spaces are named after David Hilbert , who developed this idea through his studies of integral equations. We define our valid norm using the innerproduct as:

Examples of hilbert spaces

Below we will list a few examples of Hilbert spaces . You can verify that these are valid inner products at home.

• For $\mathbb{C}^n$ , $$x\cdot y=y^Tx=\begin{pmatrix}\overline{y_0} & \overline{y_1} & \dots & \overline{y_{n-1}}\\ \end{pmatrix}\left(\begin{array}{c}{x}_0\\ x_1\\ ⋮\\ x_{n-1}\end{array}\right)=\sum_{i=0}^{n-1} x_i\overline{y_i}$$
• Space of finite energy complex functions: $L^2(\mathbb{R})$ $$f\cdot g=\int \,d t$$∞ ∞ f t g t
• Space of square-summable sequences: ${\ell }^2(\mathbb{Z})$ $$x\cdot y=\sum$$∞ ∞ x i y i