quality:corruption
Use in operator theory

One use of integration by parts in operator theory is that it shows that the −∆ (where ∆ is the Laplace operator) is a positive operator on L 2 {\displaystyle L^{2}} (see L space). If f {\displaystyle f} is smooth and compactly supported then, using integration by parts, we have

⟨ − Δ f, f ⟩ L 2 = − ∫ − ∞ ∞ f ″ (x) f (x) ¯ d x = − − ∞ ∞ + ∫ − ∞ ∞ f ′ (x) f ′ (x) ¯ d x = ∫ − ∞ ∞ | f ′ (x) | 2 d x ≥ 0. {\displaystyle {\begin{aligned}\langle -\Delta f,f\rangle _{L^{2}}&=-\int _{-\infty }^{\infty }f''(x){\overline {f(x)}}\,dx\\&=-\left_{-\infty }^{\infty }+\int _{-\infty }^{\infty }f'(x){\overline {f'(x)}}\,dx\\&=\int _{-\infty }^{\infty }\vert f'(x)\vert ^{2}\,dx\geq 0.\end{aligned}}}