Linear Algebra Done Right · Problem 3E.3

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · products-vector-spaces

Problem

Suppose V1,…,VmV_1, \ldots, V_m are vector spaces. Prove that L(V1×⋯×Vm,W)\mathcal{L}(V_1 \times \cdots \times V_m, W) and L(V1,W)×⋯×L(Vm,W)\mathcal{L}(V_1, W) \times \cdots \times \mathcal{L}(V_m, W) are isomorphic vector spaces. There is no assumption in the exercise above or in the two following exercises that the vector spaces are finite-dimensional.

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