Series Question

[bloodhound]

Having trouble with this.

 

 

Determind whether the following statement is true of false

 

If [math]a_k \in \mathbb{R}[/math] for each [math]k \in \mathbb{N}[/math] and [math]\sum_{k=1}^{\infty}a_k[/math] converges, then so does [math]\sum_{k=1}^{\infty}a_k^2[/math]. If you think the statement is true, prove it. If you think it's false, give an explicit counter-example.

[fourier jr]

Having trouble with this.

 

 

Determind whether the following statement is true of false

 

If [math]a_k \in \mathbb{R}[/math] for each [math]k \in \mathbb{N}[/math] and [math]\sum_{k=1}^{\infty}a_k[/math] converges' date=' then so does [math']\sum_{k=1}^{\infty}a_k^2[/math]. If you think the statement is true, prove it. If you think it's false, give an explicit counter-example.

 

counterexample: take a_n = (-1)^n / n

[e(ho0n3]

[math]a_n^2 = \frac{1}{n^2}[/math]

[math]\sum_{n=1}^{\infty} \frac{1}{n^2} = \zeta(2)[/math] which does converge.

[e(ho0n3]

I'm know the statement is true if a[k] > 0 for all k. I'm sure there is a conditionally convergent series that makes the statement false but I can't think of any.

[fourier jr]

counterexample: take a_n = (-1)^n / n

 

 

shit, now that I think about it I should make that a_n = (-1)^n / sqrt(n) . Now sum( a_n ) converges but sum( a_n )^2 diverges.

 

I'm doing a course on Hilbert spaces, Banach spaces & the l^p spaces (lower-case 'L') this fall, and I'll be doing lots of this sort of stuff. An l^2 space is the vector space of all sequences that are square-summable, which not all series are. (the set of a_n such that sum(a_n)^2 converges) l^1 is the vector space of all sequences that are absolutely-convergent, I said what l^2 is, l^3 is the vector space of all cube-summable sequences, and so on. If you think about it, l^1 is a proper subset of l^2, which is a proper subset of l^3, which is a proper subset of l^4, ... etc so we can always find counterexamples to stuff like what bloodhound has come up with.

[fourier jr]

am i just talking to myself now or what

[bloodhound]

cool. i suprisingly understood it. can u explain this bit "all sequences that are square-summable, which not all series are"

 

also can u tell us how the multiplication(scalar and vector), addition, subtraction is defined as in that vector space. and what are the inverse element for each vector in the space. what what the zero vector is defined as . cheers.

[fourier jr]

cool. i suprisingly understood it. can u explain this bit "all sequences that are square-summable, which not all series are"

I'l try rephrasing your problem: Let {a_k} be a sequence of real numbers such that \sum_{k=1}^{\infty}a_k converges, then {a_k} is square-summable. It's just analysis lingo which means that if I have a sequence, and I square all the terms & add them up in an infinite series, the series of squares converges. "Square-summable" is just a short way of saying it, same with "cube-summable" etc. As your problem shows, if a sequence is "summable" it isn't necessarily "square-summable"

 

also can u tell us how the multiplication(scalar and vector), addition, subtraction is defined as in that vector space. and what are the inverse element for each vector in the space. what what the zero vector is defined as . cheers.

A vector in R^3 looks like this: (a_1, a_2, a_3), but another way of looking at it is as a sequence of 3 real numbers since order matters with vectors. Now instead of 3 dimensions, make a vector have infinite dimensions, so it will look like this: (a_1, a_2, ....), which looks just like an infinite sequence, and you can add/subtract them componentwise, multiply them by scalars, the zero vector is the sequence which is all zeroes, & so on. If you've done a linear algebra course, there's an infinite-dimensional version of just about all the theorems about finite-dimentional vector spaces.

[Dave]

(we should be doing that next year, bloodhound, if I read my PYDC right)

[bloodhound]

I m looking forwared to studying it next year then. this years linear maths course was quite interesting for me.

[Dave]

I'm personally looking forward to Analysis III, since I did quite well on the Analysis module this time around.

[MandrakeRoot]

Banach space theory is very interesting and usefull indeed.

 

[math]l_1[/math] is indeed a proper subset of [math]l_2[/math], this is easily seen as "every absolutely summable sequence" has to be convergent to zero and hence there exists some N such that [math]|a_n|^2 \leq |a_n| \; \forall n \geq N[/math].

 

However they are not equal as [math]a = \{\frac{1}{n}\}_{n=1}^{\infty}[/math] is an element of [math]l_2[/math] that is not in [math]l_1[/math].

 

For all [math]l_p[/math], [math]l_p \subseteq c_0[/math] ([math]p \geq 1[/math] in this post).

 

In formula for real valued l_p's : [math]l_p = \{ a = \{a_n\}_{n=1}^\infty \subseteq \mathbb{R} \; : \sum_{n=1}^{\infty} |a_n|^p < \infty \}[/math]

 

This result is not true for the function space [math]L_p[/math] though.

 

Mandrake

[Dave]

Blimey.

 

What's that used for?

[fourier jr]

great, i confused Dave even more... thx for clearing up that Banach/L_p space stuff.

 

re: Banach spaces I know a Hilbert space is a certain kind of Banach space, and it's used in statistics & quantum mechanics. A guy I know told me that most of the theorems proved in Hilbert spaces were done by statisticians until the applications in quantum mechanics were found, because that's where it was used most.

[MandrakeRoot]

Most banach space and hilbert space theory is developped by analyst, and functional analyst and not statisticians or in probability theory.

In fact Hilbert and banach spaces are generalisations of spaces with a distance on it.

Indeed hilbert space theory and operator theory isused all the time in quantum mechanics, but these spaces come back all the time in mathematics too.

It is just very convenient to work with them, since they can be anything from the [math]\mathbb{R}^n[\math] to the space of all continuous functions from some space with values in [math]\mathbb{R}[\math], or whatever else. So in proving some result for a banach space, you have immediately your result for all these spaces allowing great flexability of your theorems and lemmas.

 

In short a metric space is a vector space with a distance on it, a mapping assigning a positive number to any pair of elements with some trivial properties

d(x,y) = 0 iff x = y

d(x,y) >= 0 for all x,y in the space

d(x,y) = d(y,x)

and finally d(x,y) <= d(x,z) + d(z,y)

 

such a space is called complete if every sequence that eventually sticks to itself has a limit.

 

A banach space is a vector space with a norm on it, thati s complete, a norm is a mapping assigning to every element a positive number :

norm(x) = 0 iff x = 0

norm(alpha x) = abs(alpha) norm(x) for every scalar alpha (complex or real depending if iti s a complex or real valued vector space)

norm(x + y) <= norm(x) + norm(y)

 

It is easily seen that d(x,y) = norm(x-y) will be a metric on this space, so every banach space is a complete metric space

 

Now a hilbert space is again more specific then that, it has an inner product, and indeed each hilbert space is specially a banach space.

 

If you guys want i could put the definitions more formally and more complete ?

 

Mandrake

[fourier jr]

what book did you use? I'm using Royden's in the fall, that's the one that the profs my dept like. I saw a bit of that stuff for the first time in the winter/spring when we did Fourier series. I guess the course I'm doing in the fall has Hilbert/Banach/L_p spaces with all the plumbing. There's some measure in there too, but that's off-topic here.

[MandrakeRoot]

I encountered this stuff in the first year of my math studies at the university and it was in some course material written by the professor.

Though any book on introduction to functional analysis would be surely contain all this. You could try to catch a book called "introduction to functional analysis" of J.B. Conway or "functional analysis" by W. Rudin (this is an excellent book but maybe a little hard to begin with)

 

Measure theory is sort of seperate but becomes all teh more powerfull when you combine metric spaces and measure theory !

 

The L_p's are just some examples of banach spaces, with rather pleasant properties, since banach spaces can be rather strange (in the sense that there exists spaces with really counter-intuitive properties).

 

 

Mandrake

[Dave]

I've heard of some of those things, especially in my Geometry to Groups notes. We cover metric spaces/Hilbert spaces next year in my Algebra I module.