So with no other structure it is clear that a vector space has no notion of convergence and thus infinite sums make no sense. However, suppose $V$ is an inner product space, then we have a norm which induces a topology on $V$ so we can talk about convergence. Is there then a notion of a basis which allows infinite sums?
My question is mainly motivated by statements I see by physicists in introductory quantum mechanics textbooks. For example, I have seen physicist claim that the vector space of solutions to the time independent Schroedinger equation confined to an infinite square well of width $a$ has a countable basis given by: $$\psi_n(x)=\sqrt{\frac{2}{a}}\sin\left(\frac{n\pi}{a}x\right)$$ and that every solution can be written as an infinite sum of these functions for certain constants $c_n\in \mathbb{R}$. So is this just abuse of language or is there actually a way of defining a basis that allows for such infinite sums?
Moreover, I have seen physicists talk about an uncountably infinite basis of functions parameterized by $\mathbb{R}$ where any solution to some PDE can be written as an integral over this basis for some choice of square integrable function $f$. Generally this is talked about in the context of Fourier transformations, which I am not crazy familiar with outside of what was covered in my undergraduate real analysis course. Does it also make sense to call this a basis in this context or is it again abuse of language?
Edit: I see in the tags that my first question is yes, and that it is called a Schauder Basis. So my main question is can we extend this notion to an uncountable basis when it makes sense to talk of integration.
In Hilbert spaces we have available the notion of an orthonormal or Hilbert basis, which is not a basis in the usual algebraic sense (sometimes called a Hamel basis in this context to distinguish it) but allows infinite sums which converge with respect to the norm. The $\psi_n$ are an orthonormal basis in this sense. These figure prominently in the spectral theorem for compact operators. This is not an abuse of language, just a generalization of the meaning of "basis" suitable to this context.
The uncountable stuff is formalized mathematically by the concept of a direct integral. I think most mathematicians would consider the use of the word "basis" here an abuse of language but maybe physicists don't, I don't know. This sort of thing is needed for the spectral theorem for non-compact operators.