I'm taking an introductory course in Machine Learning and we are looking at all the requirements a Vector Space must fufill to be a Hilbert Space. I'm not a very math versed person, i had to recently learn the inner product, equivalence relations, and what it even meant to use functions as vectors. I think those concepts are fine for me now but the Cauchy Criterion is where I feel completly lost. I have watched some Youtube videos on the matter, they usued a number line from 0 to 1 and points in between from which the distance in between to consecuitive points had to be less then a value epsilon. While the Cauchy Criterion made sense in such context I fail to see how it applies to a vector space of functions,discrete functions or random valued functions.
I though that the vectors that I would encounter in a Hilbert space would be functions like sinusoids or some signal's that decay and are convergent. Are all vectors on a Hilbert Space then part of a Cauchy sequence? Can, for example, sin(x) not be part of a Hilbert space because it's not part of a Cauchy sequence? Can there be elements of a cauchy sequence and other elements that are not part of a cauchy sequence? What if there are no other elements of a cauchy sequence in my vector space and just normal linear algebra vectors, if I use a Hilbert Space inner product will it still fufill the requirements for it to be a Hilbert Space?