the cartesian product of banach spaces is banach

[[concept]]

Corollary

If $B_{1}, B_{2}$ are Banach spaces, then $B_{1} \times B_{2}$ with norm

| | (b_{1}, b_{2}) | | := | | b_{1} | | + | | b_{2} | |

is a Banach space.

Proof

Exercise

just need to check all the definitions for the norm
check that Cauchy sequences in $B_{1} \times B_{2}$ consist of Cauchy sequences in each of $B_{1}$ and $B_{2}$ (similar to completeness of $R^{2}$ )

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Functional Analysis Lecture 4	2025-10-02

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