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 .
They are named after the Swedish mathematician Olof Hanner .
statement
Be . If so , then
![{\ displaystyle f, g \ in L ^ {p} (\ Omega, {\ mathcal {A}}, \ mu)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8636cf826d599fbc772789ce96611d1d0f4d4e8b)
![1 \ leq p \ leq 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/c94d35b201f4a50dd0574a25b95071e29386cde3)
![{\ displaystyle \ | f + g \ | _ {p} ^ {p} + \ | fg \ | _ {p} ^ {p} \ geq {\ big (} \ | f \ | _ {p} + \ | g \ | _ {p} {\ big)} ^ {p} + {\ big |} \ | f \ | _ {p} - \ | g \ | _ {p} {\ big |} ^ {p }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74c7af7c87b80e98d06f5c94af9e4a96a7615576)
and
-
.
If so , then the inequality symbols are reversed, that is,
becomes off .
![{\ displaystyle 2 \ leq p <\ infty}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f607b1fc3cdc8986a17e5cd1902d4f645245755b)
![\ geq](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcef7c0e95bb77a35fd1a874ca91f425215f3c26)
![\ leq](https://wikimedia.org/api/rest_v1/media/math/render/svg/440568a09c3bfdf0e1278bfa79eb137c04e94035)
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8822dbcedc8d6c412aa120faa8dccb1129ec32c)
![{\ displaystyle g = fg}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0040d86a965e217ef51bdad93bf21c29277277f)
![{\ displaystyle \ | f + g \ | _ {p} ^ {p} + \ | fg \ | _ {p} ^ {p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1040d458a06dfbf5b62f93fb09b757b96f62b4d3)
![{\ displaystyle \ | (f + g) + (fg) \ | _ {p} ^ {p} + \ | (f + g) - (fg) \ | _ {p} ^ {p} = \ | 2f \ | _ {p} ^ {p} + \ | 2g \ | _ {p} ^ {p} = 2 ^ {p} (\ | f \ | _ {p} ^ {p} + \ | g \ | _ {p} ^ {p})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/632fe6ad9cf3a550d582336b1711aacfdb7544cc)
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 .
![p = 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/d62e4100b94c1939c67f2d4b8580d26c78106c44)
Individual evidence
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↑ C. Schütt: functional analysis , page 73. Accessed June 24, 2020.