Editing Banach space (section)

====Comparison of complete metrizable vector topologies====

The [[Open mapping theorem (functional analysis)|open mapping theorem]] implies that when <math>\tau_1</math> and <math>\tau_2</math> are topologies on <math>X</math> that make both <math>(X, \tau_1)</math> and <math>(X, \tau_2)</math> into [[F-space|complete metrizable TVS]]es (for example, Banach or [[Fréchet space]]s), if one topology is [[Comparison of topologies|finer or coarser]] than the other, then they must be equal (that is, if <math>\tau_1 \subseteq \tau_2</math> or <math>\tau_2 \subseteq \tau_1</math> then <math>\tau_1 = \tau_2</math>).{{sfn|Trèves|2006|pp=166–173}}
So, for example, if <math>(X, p)</math> and <math>(X, q)</math> are Banach spaces with topologies <math>\tau_p</math> and <math>\tau_q,</math> and if one of these spaces has some open ball that is also an open subset of the other space (or, equivalently, if one of <math>p : (X, \tau_q) \to \Reals</math> or <math>q : (X, \tau_p) \to \Reals</math> is continuous), then their topologies are identical and the norms <math>p</math> and <math>q</math> are [[Equivalent norm|equivalent]].