Why isn't every Hamel basis a Schauder basis?

I seem to have tripped on the common Hamel/Schauder confusion.

If $X$ is any vector space (not necessarily finite dimension) and $B$ is a linearly independent subset that spans $X$, then $B$ is a Hamel basis for $X$.

If there exists a sequence $(e_n)$ such that for every $x \in X$ there exists a unique sequence of scalars $(\alpha_n)$ for which $\lim_{n \to \infty} || x - (\alpha_1e_1 + \cdots + \alpha_ne_n)|| = 0$, then $(e_n)$ is a Schauder basis for $X$.

So I'm tempted to think that every Hamel basis is also a Schauder basis; just extened the finite linear combination into an infinite one by adding zero coeeficients. I know I'm wrong, but what am I missing?

Solution 1:

The issue is ". . . unique sequence of scalars . . .". If $B$ is a Hamel basis, then there will be at least one such sequence of scalars, but there may be more.

Solution 2:

In terms of "just extened the finite linear combination into an infinite one by adding zero coeeficients":

Hamel basis could be defined on any linear space X. While for Schauder basis, we need some good topology defined on space X so that "limit" or "dense" makes sense. Thus in lots of spaces, you could not even define infinite sums.

When Banach space is infinite-dimensional, the cardinality of Hamel basis is actually uncountable. By contrast, Schauder basis is always a sequence, which means that it is, by definition, at most countable. I feel this could also illustrate a bit why sometimes it does not make sense to consider Schauder basis as an extension from Hamel basis - Hamel basis could be much "larger" compared with Schauder basis.

Sequence order matters for Schauder basis that is not unconditional, a permutation on the order of a Schauder basis might make it no longer a Shauder basis; however since Hamel basis only deals with finite linear combination, different order does not cause any issue.

Hamel basis always exists, assuming Axiom of Choice. But Schauder basis does not always exist. Since by definition, Schauder basis is at most countable, it requires the space to be separable in order for the closure of its span to be the whole space. But separable itself is not a guarantee for the existence of Schauder basis: Per Enflo showed in 1973 that even for some separable Banach space, there is no Schauder basis on it - yet it definitely has a Hamel basis. This is another example why it's not a good idea to say "extend a Hamel basis to Schauder basis"