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Why is it impossible or making no sense to define an adjoint operator for a non-densely defined operator?

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The adjoint is defined as

$$ \langle T^\ast u, v\rangle = \langle u, Tv\rangle. $$

This only defines $T^\ast u$ uniquely if the set of all $v$, for which this equation makes sense is dense.

But this means that $T$ has to be densely defined.

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    $\begingroup$ Why does densely defined implies that $T^*v$ is unique when it exists? $\endgroup$ Commented Aug 17, 2018 at 17:55

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