information:filtering
Formal adjoint in one variable

In the functional space of square-integrable functions on a real interval (a, b), the scalar product is defined by ⟨ f, g ⟩ = ∫ a b f (x) ¯ g (x) d x, {\displaystyle \langle f,g\rangle =\int _{a}^{b}{\overline {f(x)}}\,g(x)\,dx,}

where the line over f (x) denotes the complex conjugate of f (x). If one moreover adds the condition that f or g vanishes as x → a {\displaystyle x\to a} and x → b {\displaystyle x\to b}, one can also define the adjoint of T by T ∗ u = ∑ k = 0 n (− 1) k D k. {\displaystyle T^{*}u=\sum _{k=0}^{n}(-1)^{k}D^{k}\left.}

This formula does not explicitly depend on the definition of the scalar product. It is therefore sometimes chosen as a definition of the adjoint operator. When T ∗ {\displaystyle T^{*}} is defined according to this formula, it is called the formal adjoint of T.

A (formally) self-adjoint operator is an operator equal to its own (formal) adjoint.