If we have an infinite dimensional normed space $X$ and a finite dimensional one $Y$, is $X \bigoplus Y$ isomorphic to $X$?
I cannot conclude if it is true not get a counterexample.
I was thinking that if $dim(Y) = n$ and we take $x_1, \cdots, x_n \in X$ linearly independent, then $Y'=\text{span}\{x_1, \cdots, x_n\}$ it is clear that $Y$ is isomorphic to $Y'$ and, as $Y'$ is finite dimensional, it is complemented in $X$, so there exists a closed subset $Z$ of $X$ such that $X=Z \bigoplus Y'$. Then, if $Z$ is isomorphic to $X$ the result follows, but I don't know if that's true.
I've also tried to prove it using algebraic bases to construct an algebraic isomorphism between the spaces, but I cannot prove that is bicontinuous.
