how resolve it. Let $h:l^{2}\rightarrow \Re $ with $h(x)=\sum_{i=1}^{\infty }\frac{x_{i}}{i}$. Find a basis for $(ker(h))^{\perp}$, the set orthogonal of $(ker(h))$. Can I use closedness?
2026-04-11 19:50:39.1775937039
basis and orthogonal in Hilbert
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Let $K$ be the one-dimensional linear space spanned by $(1,1/2,1/3,\cdots)$. Then $\mbox{ker}(h)=K^{\perp}$. So $\mbox{ker}(h)^{\perp}=K^{\perp\perp}=K$ because $K$ is closed.