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Dual space

If X {\displaystyle X} is a normed space and K {\displaystyle \mathbb {K} } the underlying field (either the real or the complex numbers), the continuous dual space is the space of continuous linear maps from X {\displaystyle X} into K, {\displaystyle \mathbb {K},} or continuous linear functionals. The notation for the continuous dual is X ′ = B (X, K) {\displaystyle X^{\prime }=B(X,\mathbb {K})} in this article. Since K {\displaystyle \mathbb {K} } is a Banach space (using the absolute value as norm), the dual X ′ {\displaystyle X^{\prime }} is a Banach space, for every normed space X. {\displaystyle X.} The Dixmier–Ng theorem characterizes the dual spaces of Banach spaces.

The main tool for proving the existence of continuous linear functionals is the Hahn–Banach theorem.

Hahn–Banach theorem — Let X {\displaystyle X} be a vector space over the field K = R, C. {\displaystyle \mathbb {K} =\mathbb {R},\mathbb {C}.} Let further

Then, there exists a linear functional F: X → K {\displaystyle F:X\to \mathbb {K} } so that F | Y = f, and for all x ∈ X, Re (F (x)) ≤ p (x). {\displaystyle F{\big \vert }_{Y}=f,\quad {\text{ and }}\quad {\text{ for all }}x\in X,\ \ \operatorname {Re} (F(x))\leq p(x).}

In particular, every continuous linear functional on a subspace of a normed space can be continuously extended to the whole space, without increasing the norm of the functional. An important special case is the following: for every vector x {\displaystyle x} in a normed space X, {\displaystyle X,} there exists a continuous linear functional f {\displaystyle f} on X {\displaystyle X} such that f (x) = ‖ x ‖ X, ‖ f ‖ X ′ ≤ 1. {\displaystyle f(x)=\|x\|_{X},\quad \|f\|_{X^{\prime }}\leq 1.}

When x {\displaystyle x} is not equal to the 0 {\displaystyle \mathbf {0} } vector, the functional f {\displaystyle f} must have norm one, and is called a norming functional for x. {\displaystyle x.}

The Hahn–Banach separation theorem states that two disjoint non-empty convex sets in a real Banach space, one of them open, can be separated by a closed affine hyperplane. The open convex set lies strictly on one side of the hyperplane, the second convex set lies on the other side but may touch the hyperplane.

A subset S {\displaystyle S} in a Banach space X {\displaystyle X} is total if the linear span of S {\displaystyle S} is dense in X. {\displaystyle X.} The subset S {\displaystyle S} is total in X {\displaystyle X} if and only if the only continuous linear functional that vanishes on S {\displaystyle S} is the 0 {\displaystyle \mathbf {0} } functional: this equivalence follows from the Hahn–Banach theorem.

If X {\displaystyle X} is the direct sum of two closed linear subspaces M {\displaystyle M} and N, {\displaystyle N,} then the dual X ′ {\displaystyle X^{\prime }} of X {\displaystyle X} is isomorphic to the direct sum of the duals of M {\displaystyle M} and N. {\displaystyle N.} If M {\displaystyle M} is a closed linear subspace in X, {\displaystyle X,} one can associate the orthogonal of M {\displaystyle M} in the dual, M ⊥ = { x ′ ∈ X: x ′ (m) = 0, for all m ∈ M }. {\displaystyle M^{\bot }=\left\{x^{\prime }\in X:x^{\prime }(m)=0,\ {\text{ for all }}m\in M\right\}.}

The orthogonal M ⊥ {\displaystyle M^{\bot }} is a closed linear subspace of the dual. The dual of M {\displaystyle M} is isometrically isomorphic to X ′ / M ⊥. {\displaystyle X'/M^{\bot }.} The dual of X / M {\displaystyle X/M} is isometrically isomorphic to M ⊥. {\displaystyle M^{\bot }.}

The dual of a separable Banach space need not be separable, but:

Theorem — Let X {\displaystyle X} be a normed space. If X ′ {\displaystyle X'} is separable, then X {\displaystyle X} is separable.

When X ′ {\displaystyle X'} is separable, the above criterion for totality can be used for proving the existence of a countable total subset in X. {\displaystyle X.}