Definition:Hilbert Space

From ProofWiki
Jump to navigation Jump to search

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.



This page has been identified as a candidate for refactoring.
The following need to be established by the technique of equivalent definitions with an 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 $\mathsf{Pr} \infty \mathsf{fWiki}$.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Refactor}} from the code.


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 $\struct {V, \norm {\,\cdot\,}_V}$


This page has been identified as a candidate for refactoring.
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 $\mathsf{Pr} \infty \mathsf{fWiki}$.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Refactor}} from the code.


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.


This page has been identified as a candidate for refactoring.
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 $\mathsf{Pr} \infty \mathsf{fWiki}$.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Refactor}} from the code.


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

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Hilbert_Space&oldid=527485"