I had the following little question pop up, but I cannot seem to find any reference to it.

Let $X$ be a Banach space and $E\subseteq X$ a proper subspace with $E$ isomorphic to $X$ itself. Is the subspace $E$ necessarily complemented in $X$?

Composing the isomorphism with the inclusion map is a linear surjection of $X$ onto $E$, but this map is not necessarily an idempotent on $X$. So this map will require some tweaking to become a projection if it is actually the case that such subspaces are always complemented. It is not entirely clear if this can always be done, so the answer might be no in general. Is there an example?

There are some (non-Hilbert space) examples for which the answer is yes. Gowers' solution to the Hyper plane problem [1] is one, but just because that space is not isomorphic to any of its proper subspaces. Another example is one from [2] where all subspaces $E$ that are isomorphic to $X$ are *exactly* those of finite codimension, and hence are complemented.

[1] Gowers, W.T., *A Solution to Banach's Hyperplane Problem*. Bulletin of the London Mathematical Society, Volume 26, Issue 6, November 1994, Pages 523–530.

[2] Gowers, W. & Maurey, B. *Banach spaces with small spaces of operators* Math Ann (1997) 307: 543.