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:
- 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 .