It is often said that the inner product is a generalized version of the dot product. But I noticed that for the dot product, summation is done up to the dimension of the vector space.
$\sum\limits_{i=1}^{n}{a_ie_i}$ where n is the dimensionality of the basis. But in case of the inner product in a Hilbert space, the summation has nothing to do with the dimension of the space which is typically infinite. It is just an integral $\int\limits_a^b{f(x)g(x)}$ of a function f(x) and a basis function g(x) over some interval [a,b]. Or in a discrete form, it could also be written a summation $\sum\limits_{x=a}^{b}{f(x)g(x)}$. But it has nothing to do with the dimension of the space. So the dot product and the inner product are quite different, aren't they?