Recent content by Funky1981

Let X be a real Banach Space, C be a closed convex subset of X. Define Lc = {f: f - a ∈ X* for some real number a and f(x) ≥ 0 for all x ∈ C} (X* is the dual space of X) Using a version of the Hahn - Banach Theorem to show that C = ∩ {x ∈ X: f(x) ≥ 0} with the index f ∈ Lc under the...

Homework Statement Let L(l^2,l^2) be the space of bounded linear operators K:l^2->l^2. Now I define a map from l^infinite to L(l^2,l^2) as a->Ta(ei) to be Ta(ei)=aiei where ei is the orthonormal basic of l^2 and a=(a1,a2,...) is in l^infinte I want to prove this map is bijection can...

Thanks for your suggestion

Homework Statement Let fn be a sequence of measurable functions converges to f a.e. Is it possible to get a subsequence fnk of fn s.t. fn converges in L1 ? 2. The attempt at a solution I have proved the converse statement is true and i guess the above statement is impossible but I fail...

Homework Statement Let (R,M,m) be Lebesgue measure space in R. Given E contained in R with m(E)>0 show that the set E-E defined by E-E:={x in R s.t. exists a, b in E with x= a-b } contains an interval centered at the origin Homework Equations try to prove by contradiction and use...

So if I take A to be 0<y<x<1, B to be 0<x≤y<1 and construct h = f\chi_A + g\chi_B consider set {x| h(x)> a} if it is in A then we have h= f then f measurable implies h measurable, is my construction right?

Homework Statement Let f : (0,1) —>R be measurable( w.r.t. Lebesgue measure) function in L1((0,1)). Define the function g on (0,1)× (0,1) by g(x,y)=f(x)/x if 0<y<x<1 g(x,y)=0 if 0<x≤y<1 Prove: 1) g is measurable function (w.r.t. Lebesgue measure in the prodcut (0,1)× (0,1) 2)g is integrable...

1/x> 0 for all x >0 then so the measure should be infinte. but now why lim λm({x|f(x)>λ}) exists and is finite

i would write it into union of ##\{x~\vert~1/x>1/n}##??

Thanks for your example. But I cannot convince myself to understand the measure of your case here. Is the measure m({x|1/x>λ}) equal to 0 ??

If f be a measurable function. Assume that lim λm({x|f(x)>λ}) exists and is finite as λ tends to infinite Does this imply that ∫|f|dm is finite? Here m is the Lebesgue measure in R If not can anyone give me an example??

Let T: l^2 -> l^2 be bounded linear operators. K=L(l^2,l^2) be the space of T, Prove that K=L(l^2,l^2) is not separable I know that if a space contains an uncountable number of non intersecting open balls then it is not separable. But how can I apply this statement here ( I mean how to...

Homework Statement p prime, If p=1 ( mod 3) then Zp contains primitive cube roots of unity. Now I am considering which p does Zp contains primitive fourth roots of unity. opposite way? I mean if p=1(mod4) then Zp contains primitive fourth roots of unity?? 2. The attempt at a solution I...

Homework Statement Solve the following : a) Show that ordp((p^n)!)=1+p+p^2+p^3+...+p^(n-1) b)For which values of p does the following series converge in Qp? 1)1+(15/7)+(15/7)^2+(15/7)^3+... 2)1!+2!+3!+4!+... 2. The attempt at a solution For a) I want to to count how...

Thank u so much!