Clarkson's inequalities

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

Statement of the inequalities

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

:$\left\, \frac \right\, _^p + \left\, \frac \right\, _^p \le \frac \left( \, f \, _^p + \, g \, _^p \right).$

For 1 < ''p'' < 2,

:$\left\, \frac \right\, _^q + \left\, \frac \right\, _^q \le \left( \frac \, f \, _^p +\frac \, g \, _^p \right)^\frac,$

where

:$\frac1 + \frac1 = 1,$

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

:$x \mapsto x^p.$

References

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External links

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{{Functional analysis

Banach spaces
Inequalities
Measure theory