Consider [topological vector space](Topological%20vector%20spaces.md)s, $\mathcal{V}$, $\mathcal{U}$, and $\mathcal{W}$, a [binary map](Binary%20operations.md), $f:\mathcal{U}\times\mathcal{V}\rightarrow W$ is referred to as a _bilinear map_ if the corresponding [partial map](Maps.md#partial%20map)s, $f_\mathbf{u}:\mathbf{v}\rightarrow f(\mathbf{u},\mathbf{v})$ and $f_\mathbf{v}:\mathbf{u}\rightarrow f(\mathbf{u},\mathbf{v})$ are [linear](Linear%20map.md). %%this is probably more fundamental and probably applies to all sets and not topological sets.%% # Properties ## Property 1 a bilinear map, $f$ is [continuous](Continuity.md) if and only if $f$ is continuous at $(0,0).$ ## Property 2 By extension of [property 1](Bilinear%20map.md#Property%201) a _family_ $B$ of a bilinear maps is [[epicontinuous]] if and only if $B$ is epicontinuous at $(0,0).$ # Symmetry properties of bilinear maps # Examples of bilinear maps * [Commutators](Commutators.md) * [Cross products](Cross%20products.md) * [Poisson brackets](Poisson%20bracket.md) --- # Proofs and examples ## Proof of [Property 1](Bilinear%20map.md#Property%201) %%The notes that relate this to topological vector spaces are from Jaenich's topology textbook. You need a general look at bilinear maps however. See Fraleigh's book, for example %% #MathematicalFoundations/Geometry/Topology #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra