Suppose is a Hilbert space with an inner product . Consider a continuous linear operator . One can show that there exists a unique continuous linear operator with the following property:
for all .
This is a generalization of the concept of an adjoint matrix (also known as a conjugate transpose, Hermitian conjugate, or Hermitian adjoint). The adjoint of an matrix with complex entries is the matrix whose entries are defined by
where is the transpose of and is the matrix with complex conjugate entries of .
A matrix, , is said to be Hermitian if
where is the Hermitian adjoint of .
The notation used here for the adjoint, , is commonly used in linear algebraic circles (as is ). In quantum mechanics, is exclusively used for the adjoint while is interpreted as the same thing as .