I'm thinking of spaces like $$L^2[a,b] ; -\infty <a<b < \infty $$, and Schauder bases, such as that given by $$ \{\pi, Cos(n\pi), Sin(n \pi); n=1,2,...\} $$
If $$V/F$$ is a finite dimensional vector space over $F$, we can track all bases , in the sense that if $$B$$ is a basis matrix and $P$ is invertible over $F$, then $PBP^{-1}$ is also a basis for $V$. Is there any similar such thing to "keep track" of Schauder bases over infinite-dimensional Vector Spaces? Thank You,