# Documentation

Mathlib.Data.Real.ConjugateExponents

# Real conjugate exponents #

p.IsConjugateExponent q registers the fact that the real numbers p and q are > 1 and satisfy 1/p + 1/q = 1. This property shows up often in analysis, especially when dealing with L^p spaces.

We make several basic facts available through dot notation in this situation.

We also introduce p.conjugateExponent for p / (p-1). When p > 1, it is conjugate to p.

structure Real.IsConjugateExponent (p : ) (q : ) :
• one_lt : 1 < p
• inv_add_inv_conj : 1 / p + 1 / q = 1

Two real exponents p, q are conjugate if they are > 1 and satisfy the equality 1/p + 1/q = 1. This condition shows up in many theorems in analysis, notably related to L^p norms.

Instances For

The conjugate exponent of p is q = p/(p-1), so that 1/p + 1/q = 1.

Instances For
theorem Real.IsConjugateExponent.pos {p : } {q : } (h : ) :
0 < p
theorem Real.IsConjugateExponent.nonneg {p : } {q : } (h : ) :
0 p
theorem Real.IsConjugateExponent.ne_zero {p : } {q : } (h : ) :
p 0
theorem Real.IsConjugateExponent.sub_one_pos {p : } {q : } (h : ) :
0 < p - 1
theorem Real.IsConjugateExponent.sub_one_ne_zero {p : } {q : } (h : ) :
p - 1 0
theorem Real.IsConjugateExponent.one_div_pos {p : } {q : } (h : ) :
0 < 1 / p
theorem Real.IsConjugateExponent.one_div_nonneg {p : } {q : } (h : ) :
0 1 / p
theorem Real.IsConjugateExponent.one_div_ne_zero {p : } {q : } (h : ) :
1 / p 0
theorem Real.IsConjugateExponent.conj_eq {p : } {q : } (h : ) :
q = p / (p - 1)
theorem Real.IsConjugateExponent.sub_one_mul_conj {p : } {q : } (h : ) :
(p - 1) * q = p
theorem Real.IsConjugateExponent.mul_eq_add {p : } {q : } (h : ) :
p * q = p + q
theorem Real.IsConjugateExponent.symm {p : } {q : } (h : ) :
theorem Real.IsConjugateExponent.div_conj_eq_sub_one {p : } {q : } (h : ) :
p / q = p - 1
theorem Real.isConjugateExponent_iff {p : } {q : } (h : 1 < p) :
q = p / (p - 1)
theorem Real.isConjugateExponent_one_div {a : } {b : } (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) :