TITLE: Definition of Schauder basis
QUESTION [5 upvotes]: I have a definition of a Schauder basis but I’m unsure of it.
The definition I have is
A sequence $\{e_k : k \in \mathbb{N} \}$ in a normed space $(V, \| \cdot \| )$ is a Schauder basis if

$\sum_{k=1}^{\infty } \alpha _k e_k =0 $ implies $\alpha _k =0 $ for all $k$.

every $x \in V$ can be written in the form $x=\sum_{k=1}^{\infty } \alpha _k e_k $(i.e. $ \lim_{n \to \infty }\| (\sum_{k=1}^{n} \alpha _k e_k ) -x\|=0$


This definition to me seems to mean that the Schauder basis is countable.
However a theorem is ‘an infinite dimensional Hilbert space is separable if and only if it has a countable orthonormal basis’ .
This theorem seems to contradict that a Schauder basis has to be countable.
What’s the deal here?

REPLY [4 votes]: Perhaps to clarify some accidental "glosses" here:
In some contexts, a Schauder basis $B=\{v_\alpha:\alpha\in A\}$ for a vector space $V$ is a (finitely linearly independent) subset of $V$ such that the finite linear combinations of $B$ are dense in $V$.
There is no compulsion that the index set be countable, although if $V$ is "separable" in the sense of having a countable, dense subset, this will be the case.
Yes, since a sum of positive real numbers can be finite only if the index set is countable, in a Hilbert space the expressions of elements as infinite sums of linear combinations of basis elements can only involve countably many. :)
But that does not preclude the existence of Hilbert spaces without countable (Hilbert-space) bases. :)