Definition:Hilbert Space

Definition

Let $V$ be an inner product space over $\Bbb F \in \set {\R, \C}$.

Let $d: V \times V \to \R_{\ge 0}$ be the metric induced by the inner product norm $\norm {\,\cdot\,}_V$.

If $\struct {V, d}$ is a complete metric space, $V$ is said to be a Hilbert space.

The Hilbert space $V$ may be considered as one of the following:

The complete inner product space $\struct {V, \innerprod \cdot \cdot_V}$

The Banach space $\struct {V, \norm {\,\cdot\,}_V}$

The topological space $\struct {V, \tau_d}$ where $\tau_d$ is the topology induced by $d$

The vector space $\struct {V, +, \circ}_{\Bbb F}$

That is to say, all theorems and definitions for these types of spaces directly carry over to all Hilbert spaces.

In most of the literature, when studying a Hilbert space $\HH$, unless specified otherwise, it is understood that:

$\innerprod \cdot \cdot$ or $\innerprod \cdot \cdot_H$ denotes the inner product on $\HH$

$\norm {\,\cdot\,}$ or $\norm {\,\cdot\,}_\HH$ denotes the inner product norm on $\HH$

where the subscripts serve to emphasize the space $\HH$ when considering multiple Hilbert spaces.

Make sure to understand the precise definition of (especially) the inner product.

Furthermore, the parentheses around the argument of linear functionals and linear transformations on $\HH$ are often suppressed for brevity.

Make sure to understand which symbols denote scalars, operators and functionals, respectively.

This entry was named for David Hilbert.

The Hilbert space was one of the first attempts to generalise the Euclidean spaces $\R^n$.

Study of these objects eventually led to the development of the field of functional analysis.