Definition:Hilbert Space

Definition

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

Let $d: V \times V \to \R_{\ge 0}$ be the metric induced by the inner product norm $\left\Vert{\cdot}\right\Vert_V$.

If $\left({V, d}\right)$ 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 $\left({V, \left\langle{\cdot, \cdot}\right\rangle_V}\right)$

The Banach space $\left({V, \left\Vert{\cdot}\right\Vert_V}\right)$

The topological space $\left({V, \tau_d}\right)$ where $\tau_d$ is the topology induced by $d$

The vector space $\left({V, +, \circ}\right)_{\Bbb F}$

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 $H$, unless specified otherwise, it is understood that:

$\left\langle{\cdot, \cdot}\right\rangle$ or $\left\langle{\cdot, \cdot}\right\rangle_H$ denotes the inner product on $H$

$\left\|{\cdot}\right\|$ or $\left\|{\cdot}\right\|_H$ denotes the inner product norm on $H$

where the subscripts serve to emphasize the space $H$ 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 $H$ are often suppressed for brevity.

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

Hilbert spaces were among 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.

This entry was named for David Hilbert.