The dual space X^* of a normed space and its dual norm

Depends on

• The spaces \(\mathcal B(X,Y)\) and \(\mathcal B(X)\) of bounded linear operators
• The operator norm as the least bound and as the unit-sphere or unit-ball supremum
• Linear functionals and the algebraic dual $V^*=\mathcal L(V,F)$
• Complex norm and complex normed space

Used by

• Hilbert spaces are reflexive Corollary
• Relative dual norming, point separation, and recovery of the norm Corollary
• The norm of a vector is the supremum of |f(x)| over the dual unit ball Corollary
• A closed subspace of ell-infinity that is not complemented Counterexample
• Absolute polar in a normed dual pair Definition
• Annihilator notation and the preannihilator Definition
• Centered Borel Gaussian random element on a separable Banach space Definition
• Continuous annihilator of a linear subspace Definition
• Convex sets and continuous real-hyperplane separation in a normed space Definition
• Dentable bounded set and slice Definition
• Gateaux and Frechet derivatives of a functional Definition
• Linear coercive elliptic PDE control problem and reduced objective Definition
• The canonical evaluation map into the bidual Definition
• The negative Sobolev space H⁻¹(Ω) Definition
• The transpose of a bounded operator Definition
• The weak-star topology from finite evaluations Definition
• Weak topology on a normed space Definition
• Weak, strict, and strong separation Definition
• An explicit norming functional for the finite-dimensional maximum norm Example
• Norm-attaining functionals on a Hilbert space Example
• A Reiter net has an invariant-mean cluster point Lemma
• A separable Banach space has a countable norming family Lemma
• A topological invariant mean yields norm-approximately invariant densities Lemma
• Annihilators and preannihilators are norm closed Lemma
• Banach mean value estimate on a convex set Lemma
• Banach-valued Cauchy integral vanishes Lemma
• Bounded variation of a vector measure is a finite measure Lemma
• Dual ball as a closed subset of a product Lemma
• Evaluation defines a bounded scalar-linear map into the bidual Lemma
• Irreducible representations separate arbitrary C star algebras Lemma
• James convex-block norm-attainment criterion Lemma
• James nonreflexivity sequence separated from an annihilator Lemma
• Probability-density approximation of continuous tests and topological means Lemma
• Quantitative Bishop–Phelps support functional construction Lemma
• The elliptic Fredholm range condition is orthogonality to the adjoint kernel Lemma
• The fixed point property implies amenability Lemma
• The quotient ell-infinity/c₀ has no countable separating family Lemma
• The continuous dual, its completeness, and evaluation Remark
• The published Lp duality theorem in abstract notation Remark
• Two different Riesz representation theorems Remark

…and 19 more results.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Daniel Daners, Introduction to Functional Analysis, Definition 25.1 (standard reference, not scraped)
• Gerald Teschl, Topics in Real and Functional Analysis, Section 4.2 (standard reference, not scraped)