Question: Let ((mathcal{L},langlecdot, cdotangle)) be a Hilbert space and (mathcal{H}) some linear subspace. Show that (mathcal{H}) is a dense subset of (mathcal{L}) if, and only if,

Let $(\mathcal{L},\langle\cdot, \cdotangle)$ be a Hilbert space and $\mathcal{H}$ some linear subspace. Show that $\mathcal{H}$ is a dense subset of $\mathcal{L}$ if, and only if, the following property holds: $\forall x \in \mathcal{L}: x \perp \mathcal{H} \Longrightarrow x=0$.

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