By P.R. Halmos

ISBN-10: 0387906851

ISBN-13: 9780387906850

From the Preface: "This booklet was once written for the energetic reader. the 1st half includes difficulties, often preceded through definitions and motivation, and infrequently via corollaries and old remarks... the second one half, a really brief one, comprises hints... The 3rd half, the longest, involves options: proofs, solutions, or contructions, looking on the character of the problem....

This isn't really an creation to Hilbert house idea. a few wisdom of that topic is a prerequisite: at least, a learn of the weather of Hilbert house conception should still continue at the same time with the examining of this book."

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**Additional info for A Hilbert Space Problem Book**

**Example text**

L [:; h 'f', + ~V,2 - A(V" 1 " 21/ ~V' A'), v-A. L ! , /2 _ ! , / ~I -l- ') .. / 2 -l-'Jr- 1 ,/ -l-'Jr- 2 ). '" '" and hnally some components of the Einstem tensor ni = . A oijR. - ! , V ~), 2) lOr some constant 'Y (just ChecK that 60 CHAPTER 7 dy/dr = 0). 1) leads to the conclusion that y = 2G M, where M is the central mass and G is the gravitational constant. Therefore e- A = 1 - 2GMr- 1. Since G~ = G1 = 0, A + v is constant. Since the metric should look like the Lorentz metric for r huge, we conclude that A + v = 0.

1. _&U&_& -~ .. - II ... "'.. I':!. ,& ~ depends on the choice of orthonormal coordinates. Let G; denote the Gauss curvature of the projection S; of S into n n ""1 "":l v ff\l n 'V LV) ""1 " ff\l _ '03 LV) ""1 • Then the Gauss curvature G of S at p is n V - ~ ~ . ~ simply because the dot product of two vectors is just the sum of the products of the components. 1. Theorem. Let,) De a C sUrface m K'. ) Wlm respect to a Compactly supporcea c vecto1Jleta v on ,,) is given oy integrating v agamst me mean curvature vector: )\~({'\ ,.

1.. TTT . , . i1Ull~ • TT TI i111U. V . d,.. - U, 1, ~1 ) ~111·) . 7 a ~1 _ 2GMr)-1 .. ac = O. dr dr aT (IV') YlClU~ A~ (a constant), r"'-=h 7 (III") UT At (1 - 2GMr 1 ) -:- UT = f3 (a constant).

### A Hilbert Space Problem Book by P.R. Halmos

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