Minkowski inequality

The Minkowski inequality establishes the triangle inequality for $𝑝$ -seminorm and therefore Lebesgue space. Let $(𝑋, Σ, 𝜇)$ be a measure space and $𝑝 \in [1, \infty]$ . Then for any measurable functions $𝑓, 𝑔 : 𝑋 \to ℂ$ #m/thm/anal

‖ 𝑓 + 𝑔 ‖ 𝑝 \leq ‖ 𝑓 ‖ 𝑝 + ‖ 𝑔 ‖ 𝑝

Proof

#missing/proof For $𝑝 = 1$ ,

‖ 𝑓 + 𝑔 ‖ 1 = \int 𝑋 | 𝑓 + 𝑔 | 𝑑 𝜇 \leq \int 𝑋 (| 𝑓 | + | 𝑔 |) 𝑑 𝜇 \leq ‖ 𝑓 ‖ 1 + ‖ 𝑔 ‖ 1

as required.

For $𝑝 = \infty$ ,

‖ 𝑓 + 𝑔 ‖ \infty = i n f {𝐶 \in ℝ \geq 0 : 𝜇 ({𝑠 \in 𝑋 : | 𝑓 (𝑠) + 𝑔 (𝑠) | > 𝐶}) = 0} \leq i n f {𝐶 \in ℝ \geq 0 : 𝜇 ({𝑠 \in 𝑋 : | 𝑓 (𝑠) | + | 𝑔 (𝑠) | > 𝐶}) = 0} \leq ‖ 𝑓 ‖ \infty + ‖ 𝑔 ‖ \infty

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