the extension property, it must be complete (since it must be possible to extend the identity map from to the completion of ; that is, to the map ). Existence If is a continuous linear map from a vector subspace of into a complete Hausdorff space then there always exists a unique continuous linear extension of from to the closure of in . Consequently, it suffices to only consider maps from closed vector subspaces into complete Hausdorff spaces. Results Any locally convex space having the extension property is injective. If is an injective Banach space, then for every Banach space , every continuous linear operator from a vector subspace of into has a continuous linear extension to all of . In 1953, Alexander Grothendieck showed that any Banach space with the extension property is either finite-dimensional or else separable. {{Math theorem|name=Theorem |math_statement= Suppose that is a Banach space over the field Then the following are equivalent: is 1-injective; has the metric extension property; has the immediate 1-extension property; has the center-radius property; has the weak intersection property; is 1-complemented in any Banach space into which it is norm embedded; Whenever in norm-embedded into a Banach space then identity map can be extended to a continuous linear map of norm to ; is linearly isometric to for some compact, Hausdorff space, extremally disconnected space . (This space is unique up to homeomorphism). where if in addition, is a vector space over the real numbers then we may add to this list: has the binary intersection property; is linearly isometric to a complete Archimedean ordered vector lattice with order unit and endowed with the order unit norm. }} Examples Products of the underlying field Suppose that is a vector space over , where is either or and let be any set. Let which is the product of taken times, or equivalently, the set of all -valued functions on . Give its usual product topology, which makes it into a Hausdorff locally convex TVS. Then has the extension property. For any set the Lp space has both the extension property and the metric extension property. See also Citations References Topological vector spaces Theorems in functional analysis