Measure, Integration & Real Analysis · Problem 6B.10

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · vector-spaces-product

Problem

Suppose VV and WW are vector spaces. Define V×WV \times W by V×W={(f,g):f∈V and g∈W}.V \times W = \{(f, g) : f \in V \text{ and } g \in W\}. Define addition and scalar multiplication on V×WV \times W by (f1,g1)+(f2,g2)=(f1+f2,g1+g2)andα(f,g)=(αf,αg).(f_1, g_1) + (f_2, g_2) = (f_1 + f_2, g_1 + g_2) \quad \text{and} \quad \alpha(f, g) = (\alpha f, \alpha g). Prove that V×WV \times W is a vector space with these operations.
Intuitive explanation

The direct product V×WV \times W inherits vector space structure by performing operations coordinatewise—addition and scaling happen independently in each component. To prove it's a vector space, you need to verify all eight axioms (closure, associativity, commutativity, identity, inverses, and scalar multiplication properties) by leveraging the fact that VV and WW are already vector spaces. The key insight is that each axiom reduces to checking the same property separately in VV and in WW, which you get "for free" from their vector space structure.

Intuitive explanation

The direct product V×WV \times W inherits vector space structure by performing operations coordinatewise—addition and scaling happen independently in each component. To prove it's a vector space, you need to verify all eight axioms (closure, associativity, commutativity, identity, inverses, and scalar multiplication properties) by leveraging the fact that VV and WW are already vector spaces. The key insight is that each axiom reduces to checking the same property separately in VV and in WW, which you get "for free" from their vector space structure.

Related theory

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