Having X , a normed vector space and X* = B(X,R) its dual space ( R is the real numbers). Show that for all f contained in X*, we have that Ker f included in X is a closed subspace.
Knowing that X* is a Banach space, since R is a complete space, does that make X a Banach space too ? Needing a little help with this one.
That $\ker f$ is a linear subspace is evident. To prove that it is closed, notice that $\ker f = f^{-1}(\{0\})$, which is the preimage of the closed set $\{0\}$ under the continuous map $f$.
This has nothing to do with $X$ being or not being a Banach space.