Let $V$ be a vector space (over $\mathbb C$, but I don't think it matters), and $m: V\otimes V \to V$ a "multiplication" that is associative and commutative (but I do not demand that it is unital). Is it possible that $m$ is an *isomorphism* $V\otimes V \overset \sim \to V$? Yes: $V$ can be zero-dimensional, or $V$ can be one-dimensional and $m$ non-zero.


Since $0$ and $1$ are the only finite solutions to $v^2 = v$, any other example must have $\dim V = \infty$. But there are many $\infty$s and many possible maps, and although I am sure that there are some examples, I am having trouble writing one down. Hence:



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> What is an example of an infinite-dimensional vector space $V$ and an isomorphism $m: V\otimes V \overset\sim\to V$ that is associative and commutative? Or is this impossible?
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Can a commutative, associative "multiplication" on an infinite-dimensional vector space be an isomorphism?
It is impossible. Let $x$ and $y$ be linearly independent vectors in $V$. Then $x \otimes y \neq y \otimes x$ but $m(x \otimes y) = m(y \otimes x)$.


