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?)
[Math] Separable Banach Spaces vs. Non-separable ones
banach-spacesfunctional-analysisseparable-spaces
Best Answer
Separability is a topological notion. It's usefulness arises from each of the two parts of its definition:
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.
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:
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.
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.