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Kernels and image sets for an operator and its dual

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Let $\mathcal{X}$ and $\mathcal{Y}$ be Hilbert spaces. Let $A: \mathcal{X} \rightarrow \mathcal{Y}$ be a bounded and linear operator. ThenLet $\mathcal{X}$ and $\mathcal{Y}$ be Hilbert s




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