Dual Space
Every topological vector space has a continuous dual space—the set V* of all continuous linear functionals, i.e. continuous linear maps from the space into the base field K. A topology on the dual can be defined to be the coarsest topology such that the dual pairing V* × V → K is continuous. This turns the dual into a locally convex topological vector space. This topology is called the weak-* topology. This may not be the only natural topology on the dual space; for instance, the dual of a Banach space has a natural norm defined on it. However, it is very important in applications because of its compactness properties (see Banach–Alaoglu theorem).
Read more about this topic: Topological Vector Space
Famous quotes containing the words dual and/or space:
“Thee for my recitative,
Thee in the driving storm even as now, the snow, the winter-day
declining,
Thee in thy panoply, thy measur’d dual throbbing and thy beat
convulsive,
Thy black cylindric body, golden brass and silvery steel,”
—Walt Whitman (1819–1892)
“Shall we now
Contaminate our fingers with base bribes,
And sell the mighty space of our large honors
For so much trash as may be grasped thus?
I had rather be a dog and bay the moon
Than such a Roman.”
—William Shakespeare (1564–1616)