let $H$ be a separable hilbert space of infinite dimension and let $K$ a closed hyperplan . I want to show that $H$ is homeomorphic to $K$ .
2026-03-26 01:34:24.1774488864
a separable hilbert space and closed hyperplan
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$K$ is an infinite dimensional separable Hilbert space, endowed with the restriction of the scalar product of $H$. The completeness follows from the fact that $K$ is closed.
There exists a bijective isometry between any infinite dimensional separable Hilbert space and $l^2$. Therefore there exists a bijective isometry between $H$ and $K$.