Partial-sum projections and basis constant

Let $(e_n{)}_{n\ge 1}$ be a Schauder basis of $X$, with its algebraic coordinate functionals $e_n^{\ast }$. For $N\ge 1$ define the partial-sum projection

$P_{N}x:={\sum }_{n=1}^N{e}_n^{\ast }(x)e_n,$

and put $P_0:=0$. Each $P_N$ is linear, has finite-dimensional range, satisfies $P_N^2=P_N$, and obeys $P_M{P}_N=P_{\min \{M,N\}}$; these assertions use only uniqueness of coefficients.

If every $P_N$ is bounded and the real set $\{\mathrm{\parallel }{P}_N\mathrm{\parallel }:N\ge 0\}$ is bounded above, the basis constant is

$K:={\sup }_{N\ge 0}\mathrm{\parallel }{P}_N\mathrm{\parallel }\mathrm{.}$

This is an ordinary finite real supremum. Under its stated Dependent Choice hypothesis, the later boundedness theorem proves both hypotheses for every Schauder basis; until then neither $\mathrm{\parallel }{P}_N\mathrm{\parallel }$ nor $K$ is used for an algebraic projection not yet known to be bounded.