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Vector Space Polar


Given a topological vector space X and a neighborhood V of 0 in X, the polar K=K(V) of V is defined to be the set

 K(V)={Lambda in X^*:|Lambdax|<=1 for every x in V}

and where |Lambdax| denotes the magnitude of the scalar Lambdax in the underlying scalar field of X (i.e., the absolute value of Lambdax if X is a real vector space or its complex modulus if X is a complex vector space) and where X^* denotes the continuous dual space of X (i.e., X^* is the space of all continuous linear functionals from X to the underlying scalar field of X).

Worth noting is that the polar K(V) is essentially the norm unit ball in X^* and is fundamental in functional analysis, e.g., in the Banach-Alaoglu theorem which says K(V) is weak-* compact for all neighborhoods V of 0 in X.


See also

Banach-Alaoglu Theorem, Dual Vector Space, Unit Ball

This entry contributed by Christopher Stover

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References

Rudin, W. Functional Analysis. New York: McGraw-Hill, 1991.

Cite this as:

Stover, Christopher. "Vector Space Polar." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/VectorSpacePolar.html

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