The triangle inequality for L^p spaces: it bounds the L^p norm of a sum by the sum of the L^p norms, establishing the norm property for 1 ≤ p ≤ ∞ and hence the metric and linear structure of L^p spaces.

Principle

For 1 ≤ p ≤ ∞ and f,g in L^p, ||f+g||_p ≤ ||f||_p + ||g||_p. The inequality follows from convexity of t ↦ |t|^p (for p≥1) and can be proved using Hölder's inequality applied to appropriate auxiliary functions.

Demonstration

Example: in ℓ^p sequences, Minkowski states (∑ |a_n + b_n|^p)^{1/p} ≤ (∑ |a_n|^p)^{1/p} + (∑ |b_n|^p)^{1/p}. For p=2 this is the triangle inequality for Euclidean norm on sequences, directly verifiable by parallelogram-type arguments or Hölder/Cauchy–Schwarz.

Misapplication

Applying Minkowski in the quasi-norm regime 0

Consequence

Minkowski endows L^p with a norm (for p≥1), so L^p becomes a normed linear space and, with completeness, a Banach space; it underpins geometrical and functional-analytic results about convergence and continuity in these spaces.

Reversal

The reversed triangle inequality ||f||_p - ||g||_p ≤ ||f-g||_p gives a complementary bound, and equality cases in Minkowski characterize collinearity or proportionality of functions in normed-space terms.

Boundary

Requires p≥1 to hold as stated; the inequality does not apply to p<1 and presupposes measurability and finiteness of norms involved (finite L^p norm values).

Semantic Tension

Tension exists between Minkowski and quasi-normed spaces where the triangle inequality weakens; also between Minkowski and Hölder since proofs often use Hölder and both interact in L^p geometry.

Synthesis

Minkowski's inequality asserts that the L^p norm satisfies the triangle inequality for p≥1, thereby making L^p a normed (and typically complete) vector space and supplying the metric framework for analysis.