We can assume some familiarity of Matrix and Determinants. Definitely in xy plane rotation is a Linear Mapping.z-axis will be the axis of rotaion.
x' = xcosø-ysinø and y'=ycosø-xsinø and z' = z - This can be expressed as a matrix(of Linear Transformation).(Rotaion by an angle ø in xy plane...
See the phrase "is countable as countable union of finite sets". In the step above that, epsilon in the right side
should be replaced by 1/n, I think. Union on the right side should be over n. (I am not sure).
Thank you very much for the proof. But how I was supposed to prove this while...
Your proof is tough!
Looks like there are some small mistakes in the proof. For example "for all i" should be replaced by "for all j".
one "> epsilon" should be replaced by "> 1/n".
One major doubt is that in the proof it is assumed that measure of A is finite.
Consider my example V =...
"Real And Complex Analysis" by Walter Rudin 3rd Edition, Theorem 2.20 Page No:50, 51,52
The sentence troubling me is "If range of T is a subspace Y of lower dimension, Then m(Y) = 0..."
The context is that I am reading the proof that Lebesgue measure is rotation invariant
Let X be a k-dimensional euclidean space. T is a linear map and its range is a subspace Y of lower
dimension. I want to prove that m(Y) = 0 where m is the lebesgue measure in X.
How to prove this...
OK. Looks like I got the answer. It was a trivial case. If there is no such n then nw is a member of alpha for every n by induction. Since aplpha is not equal to Q and Q is archemedian it is not possible.If q is not a member
of alpha then q > any member of alpha and nw > q for some n.And nw...
Let α be a Dedekind Cut. w a positive rational.How to prove that there exists a integer n such that nw is a member of α and (n+1)w is not a member of α, using Archemedian propoerty of Q.
Suppose p is a member of α. we can find n such that nw < p < (n+1)w. So nw is
a member of α. Further I am...
Suppose A and B are open sets in a topological Hausdorff space X.Suppose A intersection B is an empty set. Can we prove that A intersection with closure of B is also empty? Is "Hausdorff" condition necessary for that?
Please help.
I am asking the question honestly.
Equ.43 is proved for any rational number and taken as definition for any real number and author claims that
the definition is equivalent to equ.33 chapter 8(which is another definition). Author claims continuity and
monotonicity of L is required for the proof...
No. We are not using continuity of any function here. If there is no rational p < x with g(x) < f(p) < f(x), then
f(x) is not the supremum. supremum will be < or = g(x).Note that f(x) is defined as f(x) = sup( k raised to y) (y rational , y<x)(k>1).
Also continuity of L(k) is the issue. In the...
E&L
E is defined by equation 25 chapter 8 page no 178
i.e., E(z) = Sigma(n=0 to INF)( (z raised to n)/(n!))
and L is inverse of that.Why the inverse exists, why E(z) is continuous and monotonically increasing like that are proved in the book.
Finally the author defines x raised to y as...