Adjoint of a Bounded Linear Operator

Let $H$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-hilbert-space-2026a}{complex Hilbert space}, with \reftext{def:inner-product-norm-2026a}{induced norm} $\lVert\cdot\rVert$, which is a \reftext{def:complex-normed-space-2026a}{norm} on $H$ by claim 2 of \ref{lem:inner-product-norm-is-norm-2026a}. Let $S$ and $T$ be \reftext{def:bounded-linear-operator-2026a}{bounded linear operators} on $H$.

The operator $S$ is \textbf{an adjoint} of $T$ if