Norm in Sobolev Space
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From: Narek Saribekyan on 23 Dec 2009 12:40 Hi, have a look at this please. http://mathworld.wolfram.com/SobolevSpace.html http://en.wikipedia.org/wiki/Sobolev_space These definitions have different norms. Can we show that they are equivalent?
From: David C. Ullrich on 23 Dec 2009 13:34
On Wed, 23 Dec 2009 09:40:04 -0800 (PST), Narek Saribekyan <narek.saribekyan(a)gmail.com> wrote: >Hi, >have a look at this please. > >http://mathworld.wolfram.com/SobolevSpace.html >http://en.wikipedia.org/wiki/Sobolev_space > >These definitions have different norms. Can we show that they are >equivalent? Yes. In the simplest case, where we're talking about a positive integer order, the only difference is that in one place you see the sum of the L_p norms of the derivatives while in the other you see the l_p sum of the L_p norms of the derivatives. So they're equivalent if ||x||_1 = sum |x_j| and ||x||_p = (sum |x_j|^p)^(1/p) are equivalent norms on R^N. In fact any two norms on R^N are equivalent; in this special case that's easy to see. For example, it's clear that for any q we have ||x||_q <= c_q max_j |x_j| max_j |x_j| <= ||x||_q and applying that with q = 1 and q = p is enough. |