Let $A$ be a basis for the infinite dimensional space $V$. Prove that $V$ is isomorphic to the direct sum of copies of the field $F$ indexed by the set $A.$ Prove that the direct product of copies of $F$ indexed by $A$ is a vector space over $F$ and it has strictly larger dimension than the dimension of $V.$
2026-02-22 19:27:25.1771788445
Vector space isomorphic to direct sum
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This is the definition of basis.
Here use Cantor's diagonal argument.