Recent content by simpleton1

Hi, I tried to translate the question as much I could but possibly something got lost on the way. The definition of the measure is that if a group or set (the professor called it group, possibly he meant a set) is bound the measure is 0 and if its complement is bound the measure is 1...

Some more thoughts : 3. Define Xn a series in ${C}^{n}$ that weakly converges to X , Therefore for every Y in ${C}^{n}$ : $$\lim_{{k}\to{\infty}}<{X}_{k},Y>=\lim_{{k}\to{\infty}}\sum_{i=1}^{n}<{{X}_{k,i}},{Y}_{i}>=\sum_{i=1}^{n}<{{X}_{i}},{Y}_{i}>$$ We need to prove that $$...

1. a. For every group in A , its complement is also in A. Every subgroup of a bound group is also bound. Using this and the de-morgan rules show that A is closed for intersection, not and finite union. A is not closed for infinite union because infinite (even countable) union of...

Hi, Could someone post a solution to the following questions : 1. Let R be the real numbers and A a collection of all groups that are either bound or their complement is bound. a. Show that A is an Algebra. Is it a sigma algebra? b. Define measure m by m(B) = {0 , max(on B) x <...

Ok, Thanks. I got the first two questions and the first part of question 3. But the second part of 3 has still got me confused - since the number of elements in a different than 0 are finite , the sum of $\left\lVert{a}\right\rVert\infty$ over a=(1,1,1,1,...0,0,0) will not be $\infty$ but a...

Re: help for test - functional analysis\mu Hi Euge - thanks for the commentation. I did post the questions under the influence of tiredness. :D I've fixed most things I think. You are right about the definition of L in question 3. I just got too lazy to Latex it.

Hi - my professor in functional analysis posted 4 prior years tests just 4 days before the test without solutions. I'd appreciate if anyone can help send solutions for the following with the following questions : 1. $\mu$ is a sigma additive measure over sigma algebra $\Sigma$. A $\in...

Also might help me solve this is if the compliment of an immeasureable group is measureable. Or if the combination of a measureable or immeasureable function is immeasureable. Can any of these be proven?

I tried to ask the professor but he also didn't understand the question. We didn't learn about semi-groups in this course. Could you please try and solve under reasonable assumptions?

Perhaps what is missing is that in this course when they say immeasureable it means immeasureable by the lebesgue measure. I would copy the question text directly but it's not in english so I just translated it.

All it says is that E is an immeasureable group contained in (0,1).

Thanks again for your help. About 1 - I'm still not sure how it can be that the triangle inequality holds since $$(\sum_{n=0}^{\infty}{(P+Q)}^{2}\left(\frac{1}{2}\right)^{\!{n}})^{1/2}>\left\lVert{P+Q}\right\rVert$$ for polynomes. About 2 - I didn't write the question in part a...

Thank you very much! Since you've been so helpful I want to ask two additional, simpler questions (hope I got the notation right) : 1. $$\left\lVert{P}\right\rVert = (\sum_{n=0}^{\infty} {P}^{2} (\left(\frac{1}{2}\right)^{\!{n}}))^\frac{1}{2}$$ where P is a polynomial defined over...

Hi, I'm taking a course in functional analysis and having some trouble with the following questions : 1. L1(R) is the space of absolutely integrable functions on R with the norm integrate(abs(f(x)) over -inf to +inf. Define a linear operator from L1(R) to L1(R) as A(f)(x)=integrate...