I am confused about Hilbert spaces. I want to understand them better.
If I understand correctly a Hilbert space is an inner product space that has either finite or infinite dimension over real or complex numbers.
The background to this question is that I am trying to teach myself the mathematical linear algebra background to Fourier Series and Fourier Transforms.
I have read that a Hilbert Space is a vector space in $\mathbb R^\infty$ - that it is an infinite dimensional space, but I have also seen elsewhere references to 'finite Hilbert Space'. This makes me think that perhaps a Hilbert space is naturally assumed to be infinite unless qualified otherwise.
Looking more closely it seems to me that
- all real inner product spaces are Hilbert spaces
- all complex inner product spaces are Hilbert spaces provided that they follow the criteria listed, for example, on the wikipedia page under definition so that for example $<a,b>=<b,a>^*$
- If I understand correctly the term Hilbert space is useful collective term for 'well behaved' inner product spaces which include sets of sine and cosine functions, euclidean spaces etc.
My question is.. Are there any fundamentals errors in this understanding of Hilbert Spaces?
Sorry I realize that question is a bit vague.
I tried to check to see if parts this question had been asked before, but did not find it. For example, the question Dimension of a Hilbert space does not seem to address any of these particular points.