Here is the problem statement (from chapter $3$ of Axler's Linear Algebra Done Right).
Give an example of a vector space $V$ and subspaces $U_1,U_2$ of $V$ such that $U_1 \times U_2$ is isomorphic to $U_1 + U_2$, but $U_1 + U_2$ is not a direct sum.
Hint: the vector space $V$ must be infinite-dimensional.
The only infinite-dimensional vector spaces mentioned in the book up to this point are $\mathbb{F}^{\infty}$ and $P(\mathbb{R})$.
I tried to construct an example using $\mathbb{F}^{\infty}$ by letting $U_1$ be the span of the standard bases $\{e_{2n}\}$ and $U_2$ the span of $\{e_{2n-1}\}$. It seems that at least one of the $U_i$ must be infinite dimensional, or else the example could be constructed with $V$ finite dimensional.
There is an isomorphism between $U_1 \times U_2$ and $U_1 + U_2$, but $U_1 \cap U_2 = \{0\}$ so we have a direct sum, which is what we're trying to avoid. I'm not sure how to proceed from here.
Best Answer
Modify your example to make the subspaces intersect. For example, we can take $U_1 = \operatorname{span} \{ e_1 \}$ and $U_2 = \operatorname{span} \{ e_i \}_{i \in \mathbb{N}}$. Then $U_1 \cap U_2 = U_1$ so $U_1 + U_2$ is not a direct sum. In addition, $U_1 \times U_2$ is isomorphic to $U_1 + U_2 = U_2$ via the linear map sending
$$ (e_1,0) \mapsto e_1, (0,e_i) \mapsto e_{i + 1}. $$