For questions about adjoint operators in inner product spaces. For adjoint functors from category theory, use the tag (adjoint-functors).
This tag is intended for questions about adjoints of operators in inner product spaces (see the corresponding tag inner-product-space), or more generally in vector spaces equipped with a nondegenerate bilinear form. If $T : X \to Y$ is a bounded linear operator between two inner product spaces, the adjoint of $T$ is the operator $T^* \colon Y \to X$ such that: $$(\forall x\in X)(\forall y\in Y):\langle Tx, y \rangle = \langle x, T^*y \rangle.$$
If $X$ and $Y$ are complex Hilbert spaces, then $T^*$ is called the Hermitian adjoint of $T$. This notion is conceptually similar to the notion of transpose of a matrix.
This tag is not intended to be used for adjoint functors from category theory; use the tag adjoint-functors instead.