Definition of Schauder basis

Question

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

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

2. 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?

Answer 1

Yes, Schauder bases are countable. At least this is how it's defined in Linear analysis by Bollobás (p. 37) and Topics in Banach Space Theory by Albiac and Kalton (Defn 1.1.1). For a separable Hilbert space, the orthonormal basis of a Hilbert space can be taken to be a Schauder basis.

Note that if a normed space admits a Schauder basis then it is necessarily separable. Indeed, the $\mathbb{Q}$-span of the Schauder basis is a countable dense set. Thus, a non-separable Hilbert space (i.e. one whose orthonormal basis is of uncountable cardinality) does not admit a Schauder basis.

Answer 2

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. :)