Analysis MOC

Minkowski inequality

The Minkowski inequality establishes the triangle inequality for [[Lebesgue space|𝑝-seminorm]] and therefore Lebesgue space. Let (𝑋,Σ,𝜇) be a measure space and 𝑝 ∈[1,∞]. Then for any measurable functions 𝑓,𝑔 :𝑋 →ℂ anal

‖𝑓+𝑔‖𝑝≤‖𝑓‖𝑝+‖𝑔‖𝑝

Proof

proof For 𝑝 =1,

‖𝑓+𝑔‖1=∫𝑋|𝑓+𝑔|𝑑𝜇≤∫𝑋(|𝑓|+|𝑔|)𝑑𝜇≤‖𝑓‖1+‖𝑔‖1

as required.

For 𝑝 =∞,

‖𝑓+𝑔‖∞=inf{𝐶∈ℝ≥0:𝜇({𝑠∈𝑋:|𝑓(𝑠)+𝑔(𝑠)|>𝐶})=0}≤inf{𝐶∈ℝ≥0:𝜇({𝑠∈𝑋:|𝑓(𝑠)|+|𝑔(𝑠)|>𝐶})=0}≤‖𝑓‖∞+‖𝑔‖∞

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