Let be a duality of complex vector spaces and a set of matrices over .
The matrix polar of is a set of matrices over , given by^{64}
Polars of sets of matrices over are defined analogously.
We have the bipolar theorems: Let be a duality of complex vector spaces and a set of matrices over .
The matrix bipolar of a set of matrices over is therefore the smallest closed and matrix convex set which contains and 0.
The absolute matrix bipolar of a set of matrices over is therefore the smallest closed and absolutely matrix convex set which contains .
So we get a characterization of the unit balls of and for a normed space .