Let X be an infinite set. Consider the set l^p(X), where 1\leq p < +\infty, of all complex functions that satisfy the inequality
\sup \{\sum_{x\in E} |f(x)|^p: E \subset X, \;\; |E|<\aleph_0 \} < +\infty .
The function \| \|_p: l^p(X)\rightarrow \mathbb{[0,+\infty]} defined by
\| f \|_p = \sup...
I think get it now. Willard states that a topological space X is paracomact iff any cover of X has an open locally finite refinement. It does not necessarilly implies that it covers X. Munkres however states that this refinement does cover X, so U should cover K.
I seem to have stuck an obvious(?) detail in the proof of this theorem. We first show that a Hausdorff paracompact space is regular. Let X be a Hausdorff paracompact space, K be a closed subset of X , and x\in X-K . Since X is Hausdorff there exists an open cover \{ V_y: \; y\in K \} such that...
Well, he recommends both of them and he said he will be giving exercises from both of these texts, although I have a feeling that most of these will be from Jackson since it's the standard and there are more copies in the library.
So, I was trying to do a derivation of my own for the FLRW metric, since I couldn't understand the one Wald had. The spatial slice M is a connected Riemannian manifold which is everywhere isotropic. That is, in every point p\in M and unit vectors in v_1,v_2\in T_p\left(M\right) there is an...
I don't know if this is the correct section for this thread. Anyway, I'm taking a graduate course in General Relavity using Straumann's textbook. I skimmed through the pages to see his derivation of the Schwarzschild metric and it assumes knowledge of Lie groups. I've never had an abstract...
I'm just poking things. :P
I'm only curious if a similar definition can be given for the Christoffel symbols of the second kind. Also, I fail to see why the 2nd expression is not coordinate independent, since X,Y,Z are arbitrary vector fields. In the third expression I just considered some...
While you are correct by saying that the Christoffel symbols give you coordinate expressions for the covariant derivative, the Christoffel symbols are defined by the equation:
\Gamma_{\lambda\mu\nu} = \frac{1}{2}\left( g_{\mu\lambda,\nu}+g_{\nu\lambda,\mu} - g_{\mu\nu,\lambda}\right)
In...
I've been trying to come up with a oordinate free formula of Christoffel symbols. For the Christoffel symbols of the first kind it's really easy. Since
\Gamma_{\lambda\mu\nu} = \frac{1}{2}\left( g_{\mu\lambda,\nu}+g_{\nu\lambda,\mu} - g_{\mu\nu,\lambda}\right)
we can easily generalize the...
I would suggest Apostol's Mathematical Analysis book for analysis or Kolmogorov's Introductory Real Analysis. Since the latter is really cheap, I would suggest getting both of them.
No. V is an finitely dimensional real vector space over the real numbers and b is just a bilinear form. It doesn't mention semi-Riemannian or even Minkowski spaces.
Maybe the author made a mistake...