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
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.