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I'm trying to rebase my know how in linear algebra, restart from scratch to get a more formal and useful set of definitions to help me with computer programming stuff .

One of the first concepts is a vector space, which is a space with :

  • a field $F$ of scalars
  • a group of vectors $V$
  • a vector sum and a vector multiplication
    • said operations are commutative and associative
    • the sum has $0$ as identity
    • the multiplication has $1$ as identity

A vector space is also an algebraic structure .

You then use vector spaces to define Hilbert spaces .

An Hilbert space is a vector space where you can perform an inner product, in particular a scalar product is needed ( aka dot product ) .

I'm puzzled by the introduction of this scalar product and why is needed, the reported consequences of this are that :

  • you can talk about:
    • angles
    • distances
    • orthogonality of lines
  • limits exist, and this is good so you can apply calculus to Hilbert spaces
  • and because limits exist ( I guess that this means that everything always converge to a point ) you can pretty much always express an Hilbert space ( or elements of an Hilbert space (?) ) with a series, a series of elements from your field ( numbers ) or a series of functions

It's not explicitly stated in any resource I found but I think that Hilbert spaces are not algebraic structure as vector spaces are, you lose this property .

Assuming that I haven't made any mistakes, is this a list of consequences from just having a scalar product defined for a vector space ?

What is a vector space without a scalar product defined ? What's a vector space that is not an Hilbert space ? How it looks like ?

I'm really having some troubles in visualizing this with some geometries or just some graphs .

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Each Hilbert space $(X, <\cdot,\cdot>)$ is also a normed space $(X, ||\cdot||)$ with norm $||x|| := \sqrt {<x,x>}, x\in X$.

Concerning the opposite direction: If $(X, ||\cdot||)$ is a normed vector space, then its norm must satisfy the parallelogram law $||x+y||^2 + ||x-y||^2=2||x||^2 + 2||y||^2$ in order to derive from an inner product.

Not every normed space is also a Hilbert space. The p-norm on $X=\mathbb R^n, n>1$, defined as $||x||_p:=(\sum_{i=1,...,n} |x_i|^p)^\frac{1}{p}, p\neq 2,$ does not derive from an inner product.

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