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.

If $\left({V, d}\right)$ is a complete metric space, $V$ is said to be a Hilbert space.

Results about Hilbert spaces are categorised in Category:Hilbert Spaces.

Standard Notation
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.

Historical Note
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.