I'm working through "Functional Analysis and Infinite-Dimensional Geometry" -Fabian et al. and I could use some help understanding one of their arguments.
Problem 1.62 asks to show that if $X$ is a normed vectore space that is topologically complete in its norm topology then it is a Banach space.
The first question is about the notion of topological completeness. I've seen this before defined as sayng a sequence $(x_n)_n$ in a topological vector space is Cauchy if for every neighbourhood of $0$ there is some N s.t. $xm-xn$ lies in that neighbourhood for all $m,n>N$, and then defining a tvs to be complete if every Cauchy sequence converges. In thale book, however, they define it as saying that X with its norm topology is (topologically) homeomorphic to a complete metric space. The question is: are these two definitions equivalent?
The second question is about Alexandrov's Theorem. Their hint says that due to Alexandrov, $X$ is a $G_{\delta}$ in its completition. I can see the conclusion following from here as one of the previous problems asks to show that any $G_{\delta}$ in a Banach space, that is also a linear subspace, must be closed. However, I've not heard of Alexandrov's Theorem before, nor was I successfull in looking it up. Does anyone know what exactly they mean by Alexandrov's Theorem?