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### Bipolar theorems

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 by64

.

The absolute matrix polar of is a set of matrices over , given by65

.

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 .

a)
[EW97b] equals its matrix bipolar if and only if is closed and matrix convex and .
b)
[EW97a] equals its absolute matrix bipolar if and only if is closed and absolutely matrix convex.

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 .

#### Footnotes

... by64
for all .
... by65
for all .

Next: Matrix extreme points Up: Matrix convexity Previous: Separation theorems   Contents   Index
Prof. Gerd Wittstock 2001-01-07