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Vx 6 £, 2G Thus a seminorm is a norm if it has the additional property that p(x) = 0 implies that x = 0. PROPOSITION 1. Consider an increasing sequence {pk}k>i of semi norms Pk defined on a vector space £. Suppose that they satisfy the condition: (3) If p^(x) = 0 for all k > I, then The function d defined by (4) Pkix - y) d(x,y)= k = l 1 + Pkix - is a distance satisfying the following conditions: (5) d{x + z, y + z) = d(x, y) y) X = 0. FRECHET SPACES 47 and for every k > 1 (6) . 11 Pk(x - y) < 2* ‘ d{x, y).

First, we shall prove inequality (28) {called the Holder inequality). -. \f{x)g{x)\dx < 1 | / | | ll/llpll0llp. ~p \ \ f Vp 1 WgKl ^ 1 P * \\gV^ P 1 P* ^, ■ TH E H O L D E R A N D TH E M IN K O W SK I IN E Q U A L IT IE S 45 It remains to show that ||/||p is a norm. If ||/|| = 0, Jl_i \f{x)\’’ dx = 0. This implies that |/(x )|^ = 0 for all x e [ —1, 1], thus that / = 0. Moreover, l|A/||p = 11/P J ' j ^ r i / ( x ) | ' ’4 x j (31) ~ f 1 “|1/P Now we must establish the inequality (32) 11/ + gWj < ll/llp + ||0||^ {called the Minkowski inequality).

0), then ||x||p = ||x||^ = 1. We have, therefore, shown (13)i. On the other hand, applying the Holder inequality for the conjugate indices r = q/p > \ and r* = q/(q — p) gives n / n \P/9/ n \iQ-p)/q (16) k=l With a,^ = \xi,\^ and \k=l / \k=l = 1, we obtain n u i l ? = I |x,|M < k=l / n \P l^ f XUJ" \k=l J " \(q-p)lQ I l \k=l = llxlljn'«-'’'/" Taking X = (1, 0 , , 0), we deduce that ||x||p = n'^"’ and ||x||, = consequently, that ||x||p/||x||, = and, ■ THE HOLDER AND THE MINKOWSKI INEQUALITIES 41 EXAMPLE 1.