Definition:Hilbert Space
Definition
Let $H$ be a vector space over $\mathbb F \in \set {\R, \C}$.
Definition 1
Let $\struct {H, \innerprod \cdot \cdot_H}$ be an inner product space.
Let $d: H \times H \to \R_{\ge 0}$ be the metric induced by the inner product norm $\norm {\,\cdot\,}_H$.
Let $\struct {H, d}$ be a complete metric space.
Then $H$ is a Hilbert space over $\mathbb F$.
Definition 2
Let $\struct {H, \norm {\,\cdot\,}_H}$ be a Banach space with norm $\norm {\,\cdot\,}_H : H \to \R_{\ge 0}$.
Let $H$ have an inner product $\innerprod \cdot \cdot_H : H \times H \to \C$ such that the inner product norm is equivalent to the norm $\norm {\,\cdot\,}_H$.
Then $H$ is a Hilbert space over $\mathbb F$.
Notation
The subscripts of the inner product $\innerprod \cdot \cdot_H$ and the inner product norm $\norm {\,\cdot\,}_H$ on $H$ serve to emphasize the space $H$ when considering multiple Hilbert spaces.
If it is clear from context which Hilbert space we are studying, the subscripts may be omitted such that::
- $\innerprod \cdot \cdot$ or $\innerprod \cdot \cdot_H$ will denote the inner product on $H$
- $\norm {\,\cdot\,}$ or $\norm {\,\cdot\,}_H$ will denote the inner product norm on $H$
Make sure to understand the precise definition of (especially) the inner product.
Also see
- Equivalence of Definitions of Hilbert Space
- Hilbert Space is Hausdorff Topological Vector Space
- Definition:Banach Space
- Definition:Linear Functional, for more discussion on notation.
- Definition:Linear Transformation, for more discussion on notation.
- Results about Hilbert spaces can be found here.
Source of Name
This entry was named for David Hilbert.
Historical Note
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.
Sources
- 1997: Reinhold Meise and Dietmar Vogt: Introduction to Functional Analysis: $\S 11$: Hilbert Spaces
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Hilbert space