Is there a normed (infinite-dimensional) space $V$ such that the only closed proper subspaces are the finite-dimensional ones?. Without the "closed" hypothesis then it is clearly false (assuming the Axiom of Choice). I think the polynomials in $[0,1]$ may what I'm looking because of the Stone-Weierstrass theorem. Does a positive answer still hold when only talking about Banach spaces?
Thanks in advance