Dual norm

In functional analysis, the dual norm is a measure of the 'size' of each continuous linear functional defined on a normed vector space. In functional analysis, the dual norm is a measure of the 'size' of each continuous linear functional defined on a normed vector space. Let X {displaystyle X} be a normed vector space with norm | ⋅ | {displaystyle |cdot |} and let X ∗ {displaystyle X^{*}} be the dual space. The dual norm of a continuous linear functional f {displaystyle f} belonging to X ∗ {displaystyle X^{*}} is defined to be the real number where sup {displaystyle sup } denotes the supremum. The map f ↦ ‖ f ‖ {displaystyle fmapsto |f|} defines a norm on X ∗ {displaystyle X^{*}} . (See Theorems 1 and 2 below.) The dual norm is a special case of the operator norm defined for each (bounded) linear map between normed vector spaces. The topology on X ∗ {displaystyle X^{*}} induced by | ⋅ | {displaystyle |cdot |} turns out to be as strong as the weak-* topology on X ∗ {displaystyle X^{*}} . If the ground field of X {displaystyle X} is complete then X ∗ {displaystyle X^{*}} is a Banach space. The double dual (or second dual) X ∗ ∗ {displaystyle X^{**}} of X {displaystyle X} is the dual of the normed vector space X ∗ {displaystyle X^{*}} . There is a natural map φ : X → X ∗ ∗ {displaystyle varphi :X o X^{**}} . Indeed, for each w ∗ {displaystyle w^{*}} in X ∗ {displaystyle X^{*}} define The map φ {displaystyle varphi } is linear, injective, and distance preserving. In particular, if X {displaystyle X} is complete (i.e. a Banach space), then φ {displaystyle varphi } is an isometry onto a closed subspace of X ∗ ∗ {displaystyle X^{**}} .