Schauder Bases

Abstract

Much of the theory of finite-dimensional normed spaces that has been presented in this book is ultimately based on Theorem 1.4.12, which says that every linear operator from a finite-dimensional normed space X into any normed space Y is bounded. A careful examination of the proof of that theorem shows that it essentially amounts to demonstrating that if x₁,…, x_n is a vector space basis for X, then each of the linear “coordinate functionals”α₁x₁ + … + α _n x _n ↦ α _m , m = 1, …, n, is bounded; this can be seen from the nature of the norm /•/ used in the proof and the argument near the end of the proof that there is no sequence (z _j ) in B _X such that $/{\text{I}}z_j / \geqslant j$ for each j. It should not be too surprising that many topological results about finite-dimensional normed spaces are ultimately based on the continuity of the members of the family $\mathfrak{B}^\#$ of coordinate functionals for some basis $\mathfrak{B}$ for the space, since it is an easy consequence of Proposition 2.4.8, Theorem 2.4.11, and the uniqueness of Hausdorff vector topologies for finite-dimensional vector spaces that the $\mathfrak{B}^\#$ norm topology of the space is the topology of the space.