In mathematics, specifically in functional analysis and Hilbert space theory, vector-valued Hahn–Banach theorems are generalizations of the Hahn–Banach theorems from linear functionals (which are always valued in the real numbers or the complex numbers ) to linear operators valued in topological vector spaces (TVSs). Definitions Throughout and will be topological vector spaces (TVSs) over the field and will denote the vector space of all continuous linear maps from to , where if and are normed spaces then we endow with its canonical operator norm. Extensions If is a vector subspace of a TVS then has the extension property from to if every continuous linear map has a continuous linear extension to all of . If and are normed spaces, then we say that has the metric extension property from to if this continuous linear extension can be chosen to have norm equal to . A TVS has the extension property from all subspaces of (to ) if for every vector subspace of , has the extension property from to . If and are normed spaces then has the metric extension property from all subspace of (to ) if for every vector subspace of , has the metric extension property from to . A TVS has the extension property if for every locally convex space and every vector subspace of , has the extension property from to . A Banach space has the metric extension property if for every Banach space and every vector subspace of , has the metric extension property from to . 1-extensions If is a vector subspace of normed space over the field then a normed space has the immediate 1-extension property from to if for every , every continuous linear map has a continuous linear extension such that . We say that has the immediate 1-extension property if has the immediate 1-extension property from to for every Banach space and every vector subspace of . Injective spaces A locally convex topological vector space is injective if for every locally convex space containing as a topological vector subspace, there exists a continuous projection from onto . A Banach space is 1-injective or a -space if for every Banach space containing as a normed vector subspace (i.e. the norm of is identical to the usual restriction to of 's norm), there exists a continuous projection from onto having norm 1. Properties In order for a TVS to have