= Adjoint on an inner product space
{title2=$\langle Tx,y\rangle=\langle x,T^*y\rangle$}
On a complex <inner product space> $V$, with the <inner product> conjugate-linear in its first argument, an everywhere-defined linear map $T:V\to V$ admits an adjoint on $V$ if there is a linear map $T^*:V\to V$ satisfying
$$
\langle Tx,y\rangle=\langle x,T^*y\rangle\qquad(x,y\in V).
$$
The adjoint, when it exists, is unique: the difference of two candidate values at $y$ is orthogonal to every vector and hence to itself. In a <finite-dimensional vector space>, it always exists, and its matrix in an <orthonormal basis> is the conjugate transpose. In a space that is not complete, existence must instead be checked.
The identity $\langle Tx,y\rangle=\langle x,Ty\rangle$ for every $x,y\in V$ supplies such an adjoint directly, namely $T^*=T$. This algebraic statement should be distinguished from <self-adjointness> of a <linear operator> on a completed <Hilbert space>, which additionally compares the domains of an operator and its Hilbert-space adjoint. The <Gaussian polynomial operator> is a useful example where the identity holds on the entire polynomial <inner product space> and can be used on every finite-dimensional invariant polynomial subspace.
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