Separable Banach Spaces vs. Non-separable ones

Question

I have just learned about separable Banach spaces. The definition of a separable space that I know is that a space is separable if you can find a countable dense subset of it. I would be appreciated if someone could point out that how really a separable and non-separable space differ. (In other words, why do we need to define separability?)

Answer 1

Separability is a topological notion. It's usefulness arises from each of the two parts of its definition:

1. Countable sets are usually easier to work with than uncountable ones.
2. Knowing the behavior of something on a dense set is often useful to prove something about it's behavior on the whole space.

We can think of separability as a way of saying that our topological space doesn't have "too many" points, because every point is just a limit point of some countable subset.

In a metric space, we can actually say more.

A metric space is separable iff it is second countable.

While we think of separability as saying that our space doesn't have "too many" points, we can think of second countability as saying that our space doesn't have "too many" open sets.

One simple result from topology that is very useful comes to mind:

If $f,g:X\to Y$ are continuous functions between topological spaces that agree on a dense subset of $X$ and $Y$ is Hausdorff, then $f=g$.

Thus knowing how a function actions on a dense set will determine it's behavior on the whole space. This type of observation allows for the construction of a metric in each of the cases of the following theorem for normed spaces.

Let $X$ be a normed space. If $X^*$ is separable, then $B_X$ is weakly metrizable. If $X$ is separable, then $B_{X^*}$ is weak* metrizable.

Combined with Banach-Alaoglu's theorem and reflexivity, we obtain results about the compactness of the weak topology. This is the Eberlein–Šmulian theorem.

Separability is also useful in the context of Banach spaces because of the notion of a Schauder basis. Only separable Banach spaces admit a Schauder basis, which allows us to write every element as a unique infinite linear combination of basis elements.

Answer 2

Separability is a property of a broader class of spaces and has several consequences. Just to mention an easy one: if your space is separable it can be seen as the union of countably many balls of any given positive radius (no matter how small). You just have to take a countable dense subset $D$ and take all the balls with such a radius and center in every $d\in D$.

Just one of many consequences.