Download Locally convex topological vector spaces Proposition: A map T:X

Locally convex topological vector spaces
Proposition: A map T:X->Y between topological spaces is
continuous if and only if for every x, every open neighborhood
W of T(x) there exists an open neigborhoud V of x such that
V is contained in T^{-1}(W).
Definition A map T:X->Y is continuous at x if every
neighborhood W of T(x) admits an open neighborhood V of x
such that V is contained in T^{-1}(W).
Proposition: Let T:X-> Y be a linear map. TFAE
i) T is continuous
ii) T is continuous at 0
iii) T is continuous at some point.
Remark: The proof is based on the following observation:
In a topological vector space the maps
S_y(x)=x+y is a homeomorphism with inverse S_{-y}.
In particular, V is neighborhood of y iff V-y is a neighborhood
at 0.
Proposition: Let f:X-> R linear and continuous. Let W be a
convex neighborhood of 0. Then either f(W)={0} or f(W) is
open.