Banach Space - Dual Space

Dual Space

If X is a Banach space and K is the underlying field (either the real or the complex numbers), then K is itself a Banach space (using the absolute value as norm) and we can define the continuous dual space as X′ = B(X, K), the space of continuous linear maps from X into K.

• Theorem If X is a normed space, then X′ is a Banach space.
• Theorem Let X be a normed space. If X′ is separable, then X is separable.

The continuous dual space can be used to define a new topology on X: the weak topology. Note that the requirement that the maps be continuous is essential; if X is infinite-dimensional, there exist linear maps which are not continuous, and therefore not bounded. The spaceX* of all linear maps into K (which is called the algebraic dual space to distinguish it from X′) also induces a weak topology which is finer than that induced by the continuous dual since X′ ⊆ X*.

There is a natural map F: X → X′′ (the dual of the dual = bidual) defined by

F(x)(f) = f(x) for all x in X and f in X′.

Because F(x) is a map from X′ to K, it is an element of X′′. The map F: x → F(x) is thus a map X → X′′. As a consequence of the Hahn–Banach theorem, this map is injective, and isometric.