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

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.