Infinite-dimensional space

Question

I've been dealing with an exercise asking to show that the infinite-dimensional space $R^\infty$ of infinite sequences is isomorphic to a proper subspace of itself. At first I thought I had to show that $R^\infty$ is isomorphic to any proper subspace of itself. But then it turned out that it was enough to present an instance of such a proper subspace, e.g. the subspace comprised by $(0,v_1,v_2,…)$, for which the isomorphism is easy to establish. So the former was a slight misinterpretation of the statement on my part. However:

Question: Is it actually at all a meaningful idea to ask oneself whether $R^\infty$ is isomorphic to any, i.e. an arbitrary, proper subspace of itself? If so, is it actually true or false?

Answer 1

It is certainly meaningful. The question you were asked had an existential quantifier, your mis-interpretation was simply to replace the existential quantifier by a universal quantifier.

However, it is evidently false. For a counterexample, take a 1-dimensional subspace.

Answer 2

Two vector spaces on the same field are isomorphic iff have the same dimension. Interesting question for you: The (infinite-dimenional) subspace of eventually zero sequences is isomorphic to the whole space?