Let be a Hilbert space and let be a densely defined linear operator.

Theorem - If $T$ is bounded then its adjoint $T^*$ is everywhere defined and is also bounded.

Proof : Since $T$ is densely defined and bounded, it extends uniquely to a bounded (everywhere defined) linear operator on , which we denote by $\widetilde{T}$ .

For each , the function defined by $f(x)=\langle \widetilde{T}x, z\rangle$ defines a bounded linear functional on . By the Riesz representation theorem there exists such that

i.e.

Since $\widetilde{T}$ extends $T$ , we also have that for every there exists such that

for every

We conclude that $T^*$ is everywhere defined. To see that it is bounded one just needs to check that

where the last inequality comes from the Cauchy-Schwarz inequality and the fact that $T$ is bounded. $\square$

Remark - This theorem shows in particular that bounded linear operators have bounded adjoints .