Dual Space - Continuous Dual Space

Continuous Dual Space

When dealing with topological vector spaces, one is typically only interested in the continuous linear functionals from the space into the base field. This gives rise to the notion of the"continuous dual space" which is a linear subspace of the algebraic dual space V*, denoted V′. For any finite-dimensional normed vector space or topological vector space, such as Euclidean n-space, the continuous dual and the algebraic dual coincide. This is however false for any infinite-dimensional normed space, as shown by the example of discontinuous linear maps.

The continuous dual V′ of a normed vector space V (e.g., a Banach space or a Hilbert space) forms a normed vector space. A norm ||φ|| of a continuous linear functional on V is defined by

 \|\varphi\| = \sup \{ |\varphi(x)| : \|x\| \le 1 \}.

This turns the continuous dual into a normed vector space, indeed into a Banach space so long as the underlying field is complete, which is often included in the definition of the normed vector space. In other words, this dual of a normed space over a complete field is necessarily complete.

The continuous dual can be used to define a new topology on V, called the weak topology.

Read more about this topic:  Dual Space

Famous quotes containing the words continuous, dual and/or space:

    If an irreducible distinction between theatre and cinema does exist, it may be this: Theatre is confined to a logical or continuous use of space. Cinema ... has access to an alogical or discontinuous use of space.
    Susan Sontag (b. 1933)

    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)

    As photographs give people an imaginary possession of a past that is unreal, they also help people to take possession of space in which they are insecure.
    Susan Sontag (b. 1933)