Uniform partial-sum criterion for a Schauder basis (${\sup }_n\Vert P_n\Vert <\infty ⟺x={\sum }_n{f}_n(x)x_n$)

For a countable minimal total family in a normed space in a Banach space, let $(x_n,f_n)$ be its unique biorthogonal system and let $P_{n}x={\sum }_{j\le n}{f}_j(x)x_j$. Uniform boundedness of these projections implies $P_{n}x\to x$ for every $x$: approximate $x$ by a finite linear combination $y$, eventually fixed by $P_n$, and bound the error by $(1+{\sup }_n\Vert P_n\Vert )\Vert x-y\Vert$. Conversely, if every vector has a norm-convergent expansion in this family, applying each continuous $f_j$ shows that its coefficients must be $f_j(x)$. Thus $P_{n}x\to x$, so the projections are pointwise bounded. The Uniform boundedness principle gives their uniform operator-norm bound. This proves the criterion and uniqueness of all expansions.