Let $T : H → H$ be a bounded self–adjoint linear operator on a Hilbert space $H$. Suppose the range $R(T)$ is dense in $H$. Prove that $T$ is injective.
2026-03-25 12:30:30.1774441830
Bounded self-adjoint linear operator is injective...
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We have $R(T)^{\perp}=\overline{R(H)}^{\perp}=H^{\perp}=\{0\}$, but $\ker T^{\ast}=(R(T))^{\perp}$.