Let $V$ be the space of all bounded functions from the set of naturals to the real numbers($f: \Bbb N \to \Bbb R$). Can we find two subspaces of $V$, say $U$ and $W$, such that there exists an isomorphism $F: U + W \to U \times W$, but this sum is not a direct sum?
As $V$ has no finite dimension, and the sum between subspaces is suppose not to be direct, one cannot relate the dimensions of the domain and the codomain to conclude the existence of such isomorphism, right? So, how to proceed?
Thanks in advance.
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Here are some hints:
Consider two subsets $A,B\subset \Bbb{N}$. Then choose $U$ as the subspace of functions $f:\Bbb{N}\to \Bbb{R}$ such that $f(k)=0$ for all $k\in\Bbb{N}\setminus A$. Choose $W$ similar but based on the subset $B$.
As the sum $U+W$ is not supposed to be a direct sum, we have to consider sets $A,B$ such that their intersection is nonempty. You also have to choose the subsets $A,B$ such that they are infinite. If this is the case, the isomorphism can be shown, but this will require some work.