## Background

This Wednesday, I read a proof about the non-separability of $\ell^\infty$ spaces. To simplify things, I assume that it’s defined on sequences.

I have written down the uncountable set (see Cantor’s diagonal argument)

in my notes. I understand

## Problem

Then, my teacher said that the reason for the non-separability of
$\ell^\infty$ was like the Pigeon-Hole Principle. I got puzzled when
I was revising the proof. In fact, in the above equation, an element
in a *dense* set $C$ can be found in each open ball $B(x,1)$. Since
the open balls $B(x,1)$ are *disjoint*, $C$ has *uncountably* many
elements. Hence, $\ell^\infty$ is non-separable.

One knows that if a space is separable, we *can’t* insert a
non-separable subspace into it. The above property of $\ell^\infty$
serves as a concrete example of this fact.

It’s easy to show that a space having a Schauder basis is separable.
Thus, we can conclude that $\ell^\infty$ *don’t* possess any Schauder
basis.

In fact, we have a more *direct* approach to the absence of Schauder
basis in $\ell^\infty$. This will be discussed in my next post.