Hilbert Spaces - Core Definitions

Hilbert spaces extend the geometric intuition of Euclidean spaces to infinite dimensions by combining the completeness of Banach spaces with the rich structure provided by an inner product.

DefinitionInner Product Space

Let $H$ be a vector space over $\mathbb{K}$ (where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$ ). An inner product on $H$ is a function $\langle \cdot, \cdot \rangle : H \times H \to \mathbb{K}$ satisfying:

Linearity in first argument: $\langle \alpha x + \beta y, z \rangle = \alpha \langle x, z \rangle + \beta \langle y, z \rangle$
Conjugate symmetry: $\langle x, y \rangle = \overline{\langle y, x \rangle}$
Positive definiteness: $\langle x, x \rangle > 0$ for all $x \neq 0$

The pair $(H, \langle \cdot, \cdot \rangle)$ is called an inner product space or pre-Hilbert space.

Every inner product induces a norm via $\|x\| = \sqrt{\langle x, x \rangle}$ . This norm satisfies the parallelogram law: $\|x + y\|^2 + \|x - y\|^2 = 2(\|x\|^2 + \|y\|^2)$ Conversely, any norm satisfying the parallelogram law arises from an inner product (the polarization identity).

DefinitionHilbert Space

A Hilbert space is an inner product space that is complete with respect to the norm induced by the inner product. That is, every Cauchy sequence converges.

ExampleStandard Examples of Hilbert Spaces

Euclidean Space: $\mathbb{R}^n$ or $\mathbb{C}^n$ with $\langle x, y \rangle = \sum_{i=1}^n x_i \overline{y_i}$
$\ell^2$ Space: The space of square-summable sequences $\ell^2 = \left\{(x_n) : \sum_{n=1}^\infty |x_n|^2 < \infty\right\}$ with inner product $\langle (x_n), (y_n) \rangle = \sum_{n=1}^\infty x_n \overline{y_n}$
$L^2$ Spaces: For a measure space $(X, \mathcal{M}, \mu)$ , $L^2(X, \mu) = \left\{f : \int_X |f|^2 \, d\mu < \infty\right\}$ with $\langle f, g \rangle = \int_X f \overline{g} \, d\mu$
Sobolev Spaces: $H^1(\Omega) = \{f \in L^2(\Omega) : \nabla f \in L^2(\Omega)\}$ with $\langle f, g \rangle = \int_\Omega (f\overline{g} + \nabla f \cdot \nabla \overline{g}) \, dx$

Remark

The inner product structure provides Hilbert spaces with geometric concepts like orthogonality, projections, and orthonormal bases. These concepts make Hilbert spaces particularly tractable compared to general Banach spaces, leading to a rich theory with applications throughout mathematics and physics.

The completeness of Hilbert spaces ensures that infinite series of orthogonal vectors converge when they should, making them the natural setting for Fourier analysis, quantum mechanics, and the study of partial differential equations.