I'm studying Functional Analysis and I'm in doubt with the definition of linear span. The book states that:
Let $\mathscr{L}$ be a topological linear space and let $\mathscr{M}$ be a linear manifold (i.e. a subset which is algebraically closed under vector sum and scalar product). Suppose the set $\mathscr{M}$ is a closed subset of $\mathscr{L}$. Then we say that $\mathscr{M}$ is a subspace of $\mathscr{L}$.
After that, the author defines the linear span of a set of vectors
Let $V$ be an arbitrary set of vectors (consisting of a finite, or countably infinite, or even uncountably infinite number of vectors). We say that $V$ spans $\mathscr{L}$ (or its subspace $\mathscr{M}$) iff $\mathscr{L}$ (or $\mathscr{M}$) is the closure of the linear manifold $\mathscr{N}$ which consists of all possible finite linear combinations of vectors from $V$.
In that case, as I understood, the linear span of $V$ is the closure of the set
$$\mathscr{N} = \left\{\sum_{k=1}^n \alpha_k x_k : x_k \in V, \ n\in \mathbb{N}\right\}.$$
After that the author says
If $V$ contains countably many vectors, the definition simplifies, and can be expressed as follows: the countable set $V = \{x_n : n \in \mathbb{N}\}$ spans $\mathscr{L}$ (or $\mathscr{M}$) if any vector $x\in \mathscr{L}$ (or $\mathscr{M}$) is either a finite linear combination $x = \sum_{k=1}^N \alpha_k x_k$ or if $x=\lim y_n$ where $y_n = \sum_{k=1}^n \alpha_k x_k$.
Now the first question: why this is the case when there are countably many elements? If the topological linear space $\mathscr{L}$ is first-countable, for example, when the topology comes from a metric, the closure should be the set of all limits of convergent sequences of points from $\mathscr{N}$. But this is related to the topology of $\mathscr{L}$, not the fact that $V$ is countable. Why the author says this is true for countable $V$?
Also, what happens in the uncountable case? In Quantum Mechanics books, it is usual to say that when we have an uncountable set of vectors $\{v_\alpha : \alpha \in A\}$ of a Hilbert space $\mathcal{H}$, their linear combination is
$$v = \int_A c(\alpha)v_\alpha d\alpha,$$
but this is quite strange. It is the integral of a function $f : A\to \mathcal{H}$ given by $f(\alpha)=c(\alpha)v_\alpha$, and I'm totally unsure on how this integral is defined. More than that I can't see connection with the definition from the book.
So why when there are countably many elements the definition simplifies like that? And when there are uncountably many elements what is the linear span, and how it relates to those "linear combinations" done in Physics? Is the definition used in Physics a special case when we work with Hilbert spaces?
I think the author means to say that things are simpler when you have a countable spanning set so that eg. you can construct orthonormal bases.
When you integrate over a continuous spectrum it does not imply that the space has uncountable dimension. Physicists sometimes loosely speak of points in the continuous spectrum as "eigenvalues" (with "eigenfunctions" that are not square integrable).