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Locally convex topological vector space
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<p>In <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a> and related areas of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of <a href="/facts/Topological_vector_space/XRex1ChN">topological vector spaces</a> (TVS) that generalize <a href="/facts/Normed_space/rmDljfyE">normed spaces</a>. They can be defined as <a href="/facts/Topological_space/v2ut7CbK">topological</a> vector spaces whose topology is <a href="/facts/Base_(topology)/8EDSDVPD">generated</a> by translations of <a href="/facts/Balanced_set/Y2ar7E6g">balanced</a>, <a href="/facts/Absorbent_set/GCtjsUuX">absorbent</a>, <a href="/facts/Convex_set/vdAuJRJl">convex sets</a>. Alternatively they can be defined as a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> with a <a href="/facts/Family_of_sets/WbYof7lP">family</a> of <a href="/facts/Seminorm/vMuescKp">seminorms</a>, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily <a href="/facts/Normable/xIbR4uE1">normable</a>, the existence of a convex <a href="/facts/Local_base/fMRM8Xht">local base</a> for the <a href="/facts/Null_vector_(vector_space)/sjkdQtdS">zero vector</a> is strong enough for the <a href="/facts/Hahn%25E2%2580%2593Banach_theorem/3AUHusvn">Hahn–Banach theorem</a> to hold, yielding a sufficiently rich theory of continuous <a href="/facts/Linear_functional/oGh7Asrz">linear functionals</a>. </p><p><a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet spaces</a> are locally convex topological vector spaces that are <a href="/facts/Completely_metrizable_space/JbJaIzI8">completely metrizable</a> (with a choice of complete metric). They are generalizations of <a href="/facts/Banach_spaces/sDEIk7EC">Banach spaces</a>, which are complete vector spaces with respect to a metric generated by a <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a>. </p>