For , the Lp norm satisfies the triangle inequality. For , apply the Holder inequality to , then divide by when it is nonzero. The cases and follow directly from the pointwise triangle inequality. In particular, a series whose term norms have a finite sum converges in the complete space.
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The Minkowski inequality is a fundamental result in the field of mathematics, specifically in the areas of functional analysis and vector spaces. It is often referred to in the context of \( L^p \) spaces, which are function spaces defined using integrable functions. The Minkowski inequality provides a means of determining the "distance" or "size" of vectors or functions in these spaces.