Why do we prefer the Schauder basis over the Hamel basis in functional analysis?

Question

Our functional analysis instructor mentioned in the class that the Hamel basis is not so important in the context of Banach spaces. Instead, we prefer the Schauder basis. However, he did not specify the reason why it is so. I tried to think about it myself, but could not find a satisfactory answer. Can someone illustrate why the Hamel basis is not important in the context of Banach spaces, and what motivated mathematicians to work with the Schauder basis?

Answer 1

Some common Banach spaces such as $\ell^p$ for $p \in [1,+\infty\rangle$ and $c_0$, the space of all sequences converging to $0$, have a simple Schauder basis consisting of canonical vectors $(e_n)_{n=1}^\infty$.

On the other hand, their algebraic dimension is equal to $c$, so a Hamel basis is necessarily uncountable and hence more complicated. Not to mention that we cannot even provide an explicit example of a Hamel basis for these spaces as its existence relies on the axiom of choice.

Answer 2

A Schauder basis captures the topology of a Banach space (or other topological vector space). It's that topology that makes the space useful and interesting.

A Hamel basis captures only the algebraic properties.

From wikipedia

In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This makes Schauder bases more suitable for the analysis of infinite-dimensional topological vector spaces including Banach spaces.