Linear functionals and the algebraic dual $V^{\ast }=\mathcal{L}(V,F)$

Depends on

• Linear map between vector spaces over the same field
• The space $\mathcal L(V,W)$ of linear maps with pointwise addition and scalar multiplication

Used by

• A type (r,s) tensor on a finite-dimensional vector space Definition
• Cotangent space and cotangent bundle as a disjoint union Definition
• Derivations at a point and the tangent space Definition
• Dual and Hom vector bundles Definition
• Local convexity, convex and balanced sets, and the continuous dual Definition
• Moment map, component Hamiltonians and infinitesimal moment maps Definition
• Symplectic and Hamiltonian Lie-group actions Definition
• The annihilator U^∘≤ V^* of U≤ V and the preannihilator ^∘ S≤ V of S≤ V^* Definition
• The canonical evaluation map J_V:V→ V^** given by J_V(v)(f)=f(v) Definition
• The coadjoint representation, action and orbits Definition
• The dual family (b^*)_b∈ B associated to a Hamel basis B, defined by b^*(c)=δ_bc Definition
• The dual or contragredient complex representation Definition
• The dual space X^* of a normed space and its dual norm Definition
• The real dominated-extension principle as an additional hypothesis over ZF Definition
• The transpose or algebraic adjoint T^*:W^*→ V^*, T^*(g)=g∘ T, of a linear map T:V→ W Definition
• A complex linear functional is recovered from its real part by f(x)=u(x)-iu(ix) Lemma
• Assuming choice, if v∉ U≤ V, some f∈ V^* vanishes on U and satisfies f(v)=1 Lemma
• The admissible values in a one-step Hahn-Banach extension form a nonempty interval Lemma
• The union of a chain of dominated extensions is a well-defined dominated linear functional Lemma
• Bilinear forms on V correspond linearly and bijectively to linear maps V→ V^* Theorem
• Finite-dimensional Riesz representation: every functional is uniquely v↦⟨ v,w⟩ Theorem
• Finite-dimensional vector spaces are rigid Theorem
• For finite-dimensional V, the canonical map V^*⊗_FW toHom_F(V,W) is an isomorphism Theorem

Dependency tree · two levels

4 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

• H. Pinkham, Linear Algebra, §6.1 (standard reference, not scraped)
• K. Conrad, Infinite-Dimensional Dual Spaces (standard reference, not scraped)