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# Hilbert space

A *Hilbert space* is an inner product space which is complete under the induced metric.

In particular, a Hilbert space is a Banach space in the norm induced by the inner product, since the norm and the inner product both induce the same metric. Any finite-dimensional inner product space is a Hilbert space, but it is worth mentioning that some authors require the space to be infinite dimensional for it to be called a Hilbert space.

Related:

InnerProductSpace, HilbertModule, QuadraticFunctionAssociatedWithALinearFunctional, VectorNorm, RieszSequence, VonNeumannAlgebra, HilbertSpacesAndQuantumGroupsVonNeumannAlgebras, L2SpacesAreHilbertSpaces, QuantumGroupsAndVonNeumannAlgebras, HAlgebra, Ries

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Definition

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Reference

## Mathematics Subject Classification

46C05*no label found*

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## Comments

## Hilbert space always Banach space?

Am I correct in thinking that a Hilbert space is always a Banach space? If so, could you say this explicitly? You can never have too many cross-references :)

## Re: Hilbert space always Banach space?

Yup. Explicitly,

"Every Hilbert space is a Banach space."

:)

Q: But why?

A: A Banach space is a normed vector space complete with respect to that norm. A Hilbert space is a special kind of this where the norm is in fact an inner product.

Cam

## Re: Hilbert space always Banach space?

Just to put it a bit differently, a Hilbert space is by definition a (particular type of) Banach space.

Any normed space whose norm comes from an inner product is called a prehilbert space. If it is complete (i.e. if it is a Banach space), then it is called a Hilbert space.

## Re: Hilbert space always Banach space?

Thanks to you both. When I said "Can you say that explicitly?", I meant "Could the author of the page please state it explicitly on the page?" PlanetMath is very useful for those of us who don't know as much as we should, and pointing out relationships helps a lot...