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We have the definition of locally convex spaces and the characterization of locally convex spaces via seminorms. We have all the ingredients to prove the very useful lemma: Let X be a vector space and P be a family of seminorms. Show that the topology induced by P is Hausdorff if and only if for every non-zero vector x ∈ X there exists a seminorm p ∈ P , such that p(x) ≠ 0.