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 following need to be established by the technique of equivalent definitions with an equivalence proof.
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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 $\struct {V, \norm {\,\cdot\,}_V}$
These can be established by the technique of iff proofs, as AFAIK you would not use them as definitions of $\HH$. If I'm wrong, then make another definition with equivalence proof
Until this has been finished, please leave {{refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
- 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.
Standard Notation
In most of the literature, when studying a Hilbert space $\HH$, unless specified otherwise, it is understood that:
- $\innerprod \cdot \cdot$ or $\innerprod \cdot \cdot_\HH$ 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.
Establish this as part of the definitions themselves.
Until this has been finished, please leave {{refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
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.
Also see
- 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
- 1990: John B. Conway: A Course in Functional Analysis ... (previous) ... (next): $\text I.1.6$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: Hilbert space
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: Hilbert space
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: Hilbert space