Clarkson's inequalities

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In mathematics, Clarkson's inequalities, named after James A. Clarkson, are results in the theory of Lp spaces. They give bounds for the Lp-norms of the sum and difference of two measurable functions in Lp in terms of the Lp-norms of those functions individually.

Statement of the inequalities[edit]

Let (X, Σ, μ) be a measure space; let fg : X → R be measurable functions in Lp. Then, for 2 ≤ p < +∞,

For 1 < p < 2,

where

i.e., q = p ⁄ (p − 1).

The case p ≥ 2 is somewhat easier to prove, being a simple application of the triangle inequality and the convexity of

References[edit]

External links[edit]