The Hahn-Banach theorem, when formulated in a topological vector space, gives rise to a number of 'separation' results. For example: Let (X,t) be a topological vector space (here X is our space, t our topology) over the real or complex field, let A be a non-empty, convex, open subset of X and M an affine subspace of X such that \(A\cap M=\emptyset\). Then there is a closed affine hyperplane H in X such that \(M\subset H\) and \(H\cap A=\emptyset.\) One purpose of the paper under review is to relax the topological conditions on the set and hyperplane mentioned in the theorem above by replacing them with their sequential analogues. When this is done the theorem becomes, as the author proves, the following: Theorem 1. Let (X,t) be a topological vector space over the real or complex field, let A be a non-empty, convex, sequentially open subset of X and M an affine subspace of X such that \(A\cap M=\emptyset\). Then there is a sequentially closed affine hyperplane H in X such that \(M\subset H\) and \(H\cap A=\emptyset.\) Another of the author's results is this: Theorem 4. Let (X,t) be a topological vector space over the real or complex field. Let A, B be two non-empty, disjoint, sequentially open, convex sets in X. Then: (1) There is a sequentially continuous not-identically-zero linear functional f on X such that f(A) and f(B) are non-empty, disjoint, open convex subsets of the field. (2) There is a sequentially closed, real affine hyperplane H in X separating A and B strictly; i.e., there is a sequentially continuous, non-identically-zero real-valued, real-linear functional f on X which separates A and B strictly. In the second section of the paper the author introduces a new class of topological vector spaces. A topological vector space (X,t) is said to be an S-locally convex space if every sequentially open, sequential neighborhood of zero in X contains a convex sequentially open, sequential neighborhood of zero; i.e., the sequential semi-topological linear space \((X,t_ s)\) generated by (X,t) is a locally convex, semi-topological linear space. Variants of this notion are discussed, interesting examples are given, and many properties of S-locally convex spaces are investigated. In the final section of the paper the general results discussed in the first section are specialized to the class of S-locally convex spaces. We shall mention only one result: Theorem 11. Let (X,t) be an S-locally convex topological vector space over the real or complex field. Let \((X,t_ s)\) be the sequential, semi- topological linear space generated by (X,t). Let A, B be two non-empty, disjoint, convex sets in X such that A is \(t_ s\)-closed and B is \(t_ s\)-compact. Then: (1) There is a sequentially continuous, not-identically-zero, linear functional f on X such that f(A) and f(B) are non-empty, convex subsets of the field, f(B) is compact, and \(\overline{f(A)}\cap f(B)=\emptyset.\) (2) There is a sequentially closed, real affine hyperplane H in X separating A and B strictly; i.e., there is a sequentially continuous, not-identically-zero, real-valued, real-linear functional f on X which separates A and B strictly.