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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.