(⇒) We assume both functions are not zero, so kf k∞ and kgk∞ > 0. n n n Combining with the Minkowski inequality, we have the desired equality. (⇐) If one of them is 0 a.e., then it’s done. (We consider the complex-valued functions here.) (b) For 1 0 such that b = λa. (a) For p = 1, since we know the triangle inequality |(f + g)(x)| ≤ |f (x)| + |g(x)| is always true, to characteriz the case when the equality holds in Minkowski’s inequality is equivalent to seeking the condition for the equality |(f + g)(x)| = |f (x)| + |g(x)| for all x, that is f (x)g(x) ≥ 0 for a.e. Real Analysis, 2nd Edition, G.B.Folland Chapter 6 Lp Spaces Yung-Hsiang Huang∗ 0 6.1 Basic Theory of Lp Spaces 1.
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