Recent content by alvielwj

I think you are using the result to prove the question . you mentioned that t is closed and bounded,then processing to prove in the way T is bounded and closed... If you using S instead of T , your prove still hold. "Now consider an open cover of T. Then add to that cover the open set V:=R\T...

Thank you. I think I need some times to understand the last step" Of that subcover, keeping only the U_i's (that is to say, removing V if it in a member of the subcover)"

I need to prove every every open cover of T has a finite subcover. My version of Heine-Borel theorem is every open cover of a closed interval has a finite subcover. It is easy to prove if it is a closed interval. I mean like [a,b] not a interval like T.

but how to prove that every open cover of T has a finite subcover

I know that I can find a cover e.g Ia=(1/n,1+1/n),union of Ia from n=1 to inifinte is a cover for S but not finite

I am studying Heine-Borel theorem. I finally get the sense of this theorem after i read a lots posts here. But I found a question posted by Rach3 why for S={1,1/2,1/3……} there is no subcollection of it is a cover of S but every open cover of T={0,1,1/2,1/3……} has a finite subcover?

Thank you for your answer.. I finally know how to use the hint.. At the beginning i really don't know how to start..

let me post the whole question Suppose that f is an entire function such that f(z) = f(z + 2π ) and f(z) = f(z + 2π i) for all z belong to C. Use Liouville's theorem to show that f is constant. Hint: Consider the restriction of f to the square {z = x + iy : 0 <x < 2π ; 0 < y <2π }

need to prove f(z) is constant. first show f is bounded,then by the Liouville's theorem, f is constant

How to show that if f is an entire function,such that f(z) = f(z + 2π ) and f(z) = f(z + 2π i) for all z belong to C. π is pi.

A question about how find the canonical forms over R and C. An example, given a quadratic form,q(x,y,z)=x^2 + 2xy + 4yz + z^2 find the canonical forms over R and C. First step,i get the matrix 1 2^0.5 0 2^0.5 0 2...

are there infinite many quadratic forms on the vector space R^n ?

For an odd prime number p let Fp be the field with p elements, ie. the integers {0...,p-1} with addition and multiplication defined modulo p. How many quadratic forms are there on the vector space Fp^n I don even know how to start this question

AH, finally,i figured it out , thank a lot

Why do we calculate the gradient vectors of the surfaces? What is the geometrical meaning of gradient vector? I think I should calculate the normals of two surfaces and get the cross product of two normals, because the vector of the intersection curve is perpendicular to both of the normals..