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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!