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Suppose V₁, Vm, and W are vector spaces. Prove that L(V₁ × ... X Vm, W) are isomorphic to L(V₁, W) × XL(Vm, W) vector spaces, where X represents the Cartesian Product. There is a shorter proof of this if one assumes that V₁, ..., Vm, and W are all finite dimensional, though one does not need to make this assumption to prove the statement.

Suppose V₁, Vm, and W are vector spaces. Prove that L(V₁ × ... X Vm, W) are isomorphic to L(V₁, W) × XL(Vm, W) vector spaces, where X represents the Cartesian Product. There is a shorter proof of this if one assumes that V₁, ..., Vm, and W are all finite dimensional, though one does not need to make this assumption to prove the statement.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Assume the vector spaces are finite for this problem. You don't need to do the problem for the vector spaces not being finite.

Suppose V₁, ..., Vm, and W are vector spaces. Prove that L(V₁ × ··· × Vm, W) are isomorphic to L(V₁, W) × × L(Vm, W) vector spaces, where × represents the Cartesian Product. There is a shorter proof of this if one assumes that V₁, ..., Vm, and W are all finite dimensional, though one does not need to make this assumption to prove the statement.
Transcribed Image Text:Suppose V₁, ..., Vm, and W are vector spaces. Prove that L(V₁ × ··· × Vm, W) are isomorphic to L(V₁, W) × × L(Vm, W) vector spaces, where × represents the Cartesian Product. There is a shorter proof of this if one assumes that V₁, ..., Vm, and W are all finite dimensional, though one does not need to make this assumption to prove the statement.
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