Schauder basis

A Schauder basis of a Banach space $X$ is a sequence $(e_n)$ such that every $x\in X$ has a unique norm-convergent expansion $x={\sum }_{n=1}^{\infty }{a}_n{e}_n$.

The nth basis projection associated with a Schauder basis is

The basis constant of a Schauder basis is ${\sup }_n\Vert P_n\Vert$, where $P_n$ are its basis projections. Uniform boundedness makes this supremum finite.

The nth coordinate functional of a Schauder basis is the bounded linear functional $e_n^{\ast }({\sum }_k{a}_k{e}_k)=a_n$.

The dual sequence consists of the coordinate functionals $(e_n^{\ast })$ in $X^{\ast }$. It is always a basic sequence, though it need not span the entire dual space.

A Schauder basis is shrinking when its dual sequence is a Schauder basis of the whole dual space. Every Schauder basis of a reflexive Banach space is shrinking.

A basic sequence in a Banach space is a sequence that is a Schauder basis for its closed linear span.

If a bounded subset $K$ of a Banach space is bounded away from zero and zero lies in its weak closure, then $K$ contains a basic sequence. More generally, weak closure may be replaced by closure in any Hausdorff locally convex vector topology weaker than the weak topology.