Orthogonal basis for infinite-dimensional vector spaces

It is quite simple to show that all finite-dimensional vector spaces with inner product have an orthogonal basis, with the standard definition of a basis from linear algebra. However, I am in trouble to find any reference about infinite-dimensional case.

Do all infinite-dimensional vector space with inner product have an orthogonal basis (again with the standard definition of a basis from linear algebra)?

From your comments (correct me if I am wrong), it appears that you are using "basis" in the following sense:

a linearly independent subset $B$ of a vector space $V$ such that every vector in $V$ may be written as a finite linear combination of vectors from $B$.

In the context of infinite dimensional vector spaces, people usually call such a set a Hamel basis.

In that case, the answer to your question is:

Some infinite-dimensional inner product spaces admit an orthogonal Hamel basis; others do not.

For one that does, consider the vector space of polynomials, equipped with the inner product $\langle p,q\rangle = \int_{-1}^1 p(x) q(x)\,dx$ (i.e. the $L^2([-1,1])$ inner product. Then the Legendre polynomials form an orthogonal Hamel basis.

For one that does not, consider any infinite-dimensional separable Hilbert space $H$ (such as $\ell^2$). It's a consequence of the Baire category theorem that any Hamel basis for $H$ is necessarily uncountable. On the other hand, because of separability, any orthogonal set is necessarily at most countable. (By rescaling we can assume all vectors in the set are unit vectors; then they are all at distance $1/\sqrt{2}$ from one another. If we put a ball of radius $1/2\sqrt{2}$ around each one, those balls are disjoint. But being separable, our space has a countable dense subset $E$, and each of these balls must contain an element of $E$.)