Inner Product Space \(\Rightarrow\) Topology
This post outlines how an inner product space can be given a canonical topological structure. I will assume that you know some basic knowledge about fields and vector/metric/topological spaces.
Maths
An "Almost Always" Approach to Sequences
Introduction
A sequence in some space \(X\) is usually represented by some function \(a:\mathbb{N}\to X\) so that the first term of the sequence is \(a(1)\), the second is \(a(2)\) and so on. However, indices are often used to represent these terms, i.e. we usually write \(a_n\) in place of \(a(n)\). Keep in mind that we shall switch between these two conventions where appropriate.
A technique for constructing subsequences
Say we are given a sequence \((a_n)\) from which we would like to construct a subsequence \((a_{n_k})\). We can think of each \(a_n\) as a fish flowing down a river, and they do so in sequence; first comes the fish \(a_1\), then \(a_2\), and so on. The idea here is to set up a "fishing net". Depending on the net, some fish can easily swim through while others may get caught. As the sequence of fish \(a_1, a_2, \ldots\) attempt to swim through the net, we note down each time a fish gets caught and that is how we construct our subsequence. Of course, in order for all of this to work we have to make sure that our "net" does not eventually run out of fish to catch.
Sums over Countable Spaces
The Finite Case
Suppose I have a space \(X\) with three elements, we can represent this space with a set \(X:=\{\alpha, \beta, \gamma\}\). I want to be able to talk about sums over this space. Of course, \(\alpha + \beta + \gamma\) doesn't make any sense. The elements of \(X\) simply serve as placeholders for the values that we want to add, in other words, we are using the "three-ness" of the space \(X\) to define a sum of three values. We can do this by assigning to each element \(x\in X\) a real number \(r\in \mathbb{R}\), i.e. by defining a weight function \(f:X\to \mathbb{R}\). For example, if we define \(f\) such that \(f(\alpha) = f(\beta) = 3\) and \(f(\gamma)=10\), then it makes sense to proceed as follows :
\[
\begin{align*}
\sum_{x\in X}f(x) &= f(\alpha) + f(\beta) + f(\gamma) \\
&= 3 + 3 + 10 \\
&= 16
\end{align*}
\]