Hanner inequalities

The Hanner inequalities come from functional analysis and are inequalities for L ^p norms . They have some important consequences, including that the L ^p spaces are for uniformly convex spaces . ${\displaystyle 1<p<\infty }$

They are named after the Swedish mathematician Olof Hanner .

statement

Be . If so , then ${\displaystyle f,g\in L^{p}(\Omega ,{\mathcal {A}},\mu )}$ ${\displaystyle 1\leq p\leq 2}$

${\displaystyle \|f+g\|_{p}^{p}+\|f-g\|_{p}^{p}\geq {\big (}\|f\|_{p}+\|g\|_{p}{\big )}^{p}+{\big |}\|f\|_{p}-\|g\|_{p}{\big |}^{p}}$

and

${\displaystyle 2^{p}{\big (}\|f\|_{p}^{p}+\|g\|_{p}^{p}{\big )}\geq {\big (}\|f+g\|_{p}+\|f-g\|_{p}{\big )}^{p}+{\big |}\|f+g\|_{p}-\|f-g\|_{p}{\big |}^{p}}$.

If so , then the inequality symbols are reversed, that is, becomes off . ${\displaystyle 2\leq p<\infty }$ ${\displaystyle \geq }$ ${\displaystyle \leq }$

Explanations of the inequalities

The second inequality is obtained from the first by substituting and . Because then the left side becomes too ${\displaystyle f=f+g}$ ${\displaystyle g=f-g}$ ${\displaystyle \|f+g\|_{p}^{p}+\|f-g\|_{p}^{p}}$

${\displaystyle \|(f+g)+(f-g)\|_{p}^{p}+\|(f+g)-(f-g)\|_{p}^{p}=\|2f\|_{p}^{p}+\|2g\|_{p}^{p}=2^{p}(\|f\|_{p}^{p}+\|g\|_{p}^{p})}$

and the right side is reshaped similarly.

For the norm is induced by a scalar product . In this case, the inequalities become equations and are equivalent to the parallelogram equation . ${\displaystyle p=2}$

Individual evidence

1. ↑ C. Schütt: functional analysis , page 73. Accessed June 24, 2020.