Download PDF by Jean-Pierre Aubin, Bernard Cornet, Herve Moulin: Applied Abstract Analysis

By Jean-Pierre Aubin, Bernard Cornet, Herve Moulin

Similar mathematics_1 books

Aimed toward the neighborhood of mathematicians engaged on usual and partial differential equations, distinction equations, and practical equations, this e-book comprises chosen papers in accordance with the displays on the foreign convention on Differential & distinction Equations and purposes (ICDDEA) 2015, devoted to the reminiscence of Professor Georg promote.

Download e-book for kindle: D.D. Kosambi: Selected Works in Mathematics and Statistics by Ramakrishna Ramaswamy

This booklet fills an enormous hole in stories on D. D. Kosambi. For the 1st time, the mathematical paintings of Kosambi is defined, gathered and provided in a way that's obtainable to non-mathematicians besides. a few his papers which are tricky to procure in those components are made on hand right here.

Additional info for Applied Abstract Analysis

Sample text

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.