DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

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The space $L (V, W)$ of linear maps with pointwise addition and scalar multiplication

Definition

Let $V$ and $W$ be vector spaces over the same field $F$ . Write

$L (V, W) : = {T : V \to W : T is linear} .$

For $S, T \in L (V, W)$ and $λ \in F$ , define pointwise operations by

$(S + T) (v) : = S (v) + T (v), (λ T) (v) : = λ T (v) .$

The zero function $v \mapsto 0_{W}$ is denoted by $0$ . Closure and the vector-space axioms for these operations are proved in $L (V, W)$ is a vector space over the common scalar field ↗.

Depends on

Used by

Adjoint representation of a Lie algebra Definition
Chevalley–Eilenberg cochains Definition
Conjugation and the adjoint representation of a Lie group Definition
Dual and Hom vector bundles Definition
Eigenvalues, eigenvectors, eigenspaces E_λ(T)=ker(T-λ I), and the spectrum σ_F(T) of an endomorphism Definition
Intertwiners, the spaces Hom_G(V,W) and End_G(V), equivalent representations, and faithful representations Definition
Linear functionals and the algebraic dual V^*=L(V,F) Definition
Polynomial evaluation at an endomorphism: p(T)=∑ₖ aₖTᵏ Definition
Representations of Lie algebras Definition
The spaces (B(X,Y)) and (B(X)) of bounded linear operators Definition
Direct-sum, dual, Hom, and tensor representations Proposition
L(V,W) is a vector space over the common scalar field Proposition
Bilinear forms on V correspond linearly and bijectively to linear maps V→ V^* Theorem

Dependency tree · two levels

9 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

S. Schiavone, MIT 18.700 Day 9, §II.3.1 (standard reference, not scraped)

The space L(V,W)L(V,W) of linear maps with pointwise addition and scalar multiplication