Mathematics

Checkpoints

• Differentiation of Elementary Functions Proof
• Hölder's Inequality Proof Proves Hölder's inequality |∑ aᵢbᵢ| ≤ (∑ |aᵢ|ᵖ)^{1/p} (∑ |bᵢ|ᵍ)^{1/q} for conjugate exponents 1/p + 1/q = 1 with p, q > 1, by normalising the sequences and applying Young's inequality termwise.
• Minkowski's Inequality Proof Proves Minkowski's inequality (∑ |aᵢ + bᵢ|ᵖ)^{1/p} ≤ (∑ |aᵢ|ᵖ)^{1/p} + (∑ |bᵢ|ᵖ)^{1/p} for p ≥ 1, the triangle inequality for the ℓᵖ norm, by splitting |aᵢ + bᵢ|ᵖ = |aᵢ + bᵢ|·|aᵢ + bᵢ|^{p−1} and applying Hölder's inequality to each piece.
• Young's Inequality Proof Proves Young's inequality ab ≤ aᵖ/p + bᵍ/q for non-negative a, b and conjugate exponents 1/p + 1/q = 1, p, q > 1, by applying the convexity of exp combined with Jensen's inequality.