Consider a Hilbert space $H_2$ with norm $\|\cdot\|_2$. Consider a linear space $H_1 \subset H_2$ so that $H_1$ is dense in $H_2$. Consider an orthonormal sequence $(h_k)_{k \in \mathbb{N}}$ in $H_2$.
Can I always find an orthogonal sequence $(g_k)_{k \in \mathbb{N}}$ in $H_2$ so that, for any $k \in \mathbb{N}$,
$$g_k\in H_1$$
and
$$\|h_k - g_k\|_2 \leq \frac{1}{1+k^2}\; ?$$
2026-03-26 01:10:26.1774487426
Approximating orthonormal basis in Hilbert space by orthonormal basis from dense subset
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