I Understanding Hilbert Vector Spaces

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Hello,

I think I understand what a vector space is. It is inhabited by objects called vectors that satisfy a certain number of properties. The vectors can be functions whose integral is not infinite, converging sequences, etc.
The vector space can be finite dimensional or infinite dimensional and be equipped with an inner product (an operation between two vectors that results in a number). That said, what is special about a Hilbert vector space compared to a non Hilbert vector space? What extra important properties does a Hilbert vector space have?

Thank you!
 

fresh_42

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Here's a short summary and answer to your question.
https://www.physicsforums.com/insights/hilbert-spaces-relatives/
At least it should allow you to be more precise on what you want to know, for otherwise a full answer to your question is a book on functional analysis.

The most important properties are, that it is either real or complex, so no other fields are considered, and that it has an inner product. It is far more a topological object than it is a linear one. The topology has many consequences. And norms are not always induced by an inner product.
 

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The main difference between a non-Hilbert space and a Hilbert space is that the latter is equipped with an inner product, while the former is not. An inner product is a very useful tool: it gives rise to notions like angles between vectors, distance (we can define a norm using an inner product, which gives rise to a metric), ... and we can start doing geometry in those spaces. Therefore, we can look at Hilbert spaces as a generalisation of Euclidean n-spaces. By definition, the space is complete with respect to the norm induced by the inner product, and hence we can do analysis on a complete space.
 

PeroK

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Hello,

I think I understand what a vector space is. It is inhabited by objects called vectors that satisfy a certain number of properties. The vectors can be functions whose integral is not infinite, converging sequences, etc.
The vector space can be finite dimensional or infinite dimensional and be equipped with an inner product (an operation between two vectors that results in a number). That said, what is special about a Hilbert vector space compared to a non Hilbert vector space? What extra important properties does a Hilbert vector space have?

Thank you!
http://mathworld.wolfram.com/HilbertSpace.html

Not that hard to find, really.

Also, to correct you on one point. The vectors, in a vector space, can be functions - full stop. The "finite integral" property is only relevant if you want to use the integral as your inner product.
 

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