Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 22I / c

The canonical embedding puts $X$ isometrically, hence with closed range, inside $X^{\ast \ast }\simeq {\ell }_2$. Every closed subspace of a Hilbert space is Hilbert and, in the infinite-dimensional separable case here, is isomorphic to ${\ell }_2$. Thus $X\simeq {\ell }_2$.