I am trying to teach myself about infinite inner product spaces.
In the article "Orthonormal basis" in Wikipedia it reads
In functional analysis, the concept of an orthonormal basis can be generalized to arbitrary (infinite-dimensional) inner product spaces. Given a pre-Hilbert space H, an orthonormal basis for H is an orthonormal set of vectors with the property that every vector in H can be written as an infinite linear combination of the vectors in the basis. In this case, the orthonormal basis is sometimes called a Hilbert basis for H. Note that an orthonormal basis in this sense is not generally a Hamel basis, since infinite linear combinations are required. Specifically, the linear span of the basis must be dense in H, but it may not be the entire space.
If we go on to Hilbert spaces, a non-orthonormal set of vectors having the same linear span as an orthonormal basis may not be a basis at all. For instance, any square-integrable function on the interval [−1, 1] can be expressed (almost everywhere) as an infinite sum of Legendre polynomials (an orthonomal basis), but not necessarily as an infinite sum of the monomials x^n.
Because I am new to all this I struggle to understand what it actually means and why that last example is true.
In the article "inner product space" under "Orthonormal sequences" the first six lines or so (up to "E is dense in V") remain a bit of a mystery to me, too.
Any insight to these slightly cryptic to me statements is appreciated.