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 : ) :

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.

  • one_lt : 1 < p
  • inv_add_inv_conj : 1 / p + 1 / q = 1
Instances For

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

    Equations
    Instances For
      theorem Real.isConjugateExponent_iff {p : } {q : } (h : 1 < p) :
      theorem Real.isConjugateExponent_one_div {a : } {b : } (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) :