I am trying to show that there exists a normed vector space which is isometric to a proper subspace of itself. I have been playing around with the $l^\infty$ norm on $\mathbb{N}$, but am struggling to find a specific mapping $f$ which maps to a subspace and is an isometry.
It's probably glaringly obvious, but I would appreciate a hint or a point in the right direction.
Consider the $\ell^2$ space of maps $\mathbb{N} \to \mathbb{R}$. Then the $\ell^2$ space of maps $\mathbb{N} \to \mathbb{R}$ which vanish on odd numbers is a proper subspace. The isomorphism comes from the bijection $2\mathbb{N} \to \mathbb{N}$.