Sequences in lp spaces (Functional Analysis)

In summary: However, if we integrate the partial sums of the sequence, we see that the RHS of the equation (ln(x+2))^p = x does not necessarily hold for all x. This suggests that there may be an infinite number of solutions to the equation (ln(x+2))^p = x. However, because the sequence (1/ln(n+1))^p is not converging, we cannot say for sure that there are an infinite number of solutions.
  • #1
just.so
8
0
[SOLVED] Sequences in lp spaces... (Functional Analysis)

Homework Statement


Find a sequence which converges to zero but is not in any lp space where 1<=p<infinity.


Homework Equations


N/A


The Attempt at a Solution


I strongly suspect 1/ln(n+1) is a solution.

Since ln(n+1) -> infinity as n -> infinity, we have 1/ln(n+1) -> 0 as n-> infinity.

I attempted using the integration test to show that partial sums for the sequence (1/ln(n+1))^p do not converge, but integrating 1/ln(n) became a bit problematic!

Attempt two was to show that each successive term in the sequence {(1/ln(n+1))^p} is larger than each successive term in some tail of the harmonic sequence.

i.e. There exists m such that
(1/ln(2))^p >= 1/(m+0) or m >= (ln(2))^p
(1/ln(3))^p >= 1/(m+1) or m >= (ln(3))^p - 1
(1/ln(4))^p >= 1/(m+2) or m >= (ln(4))^p - 2
.
.
.
In general for such an m to exist, m >= (ln(x+2))^p - x

And therefore I need to show that this right hand side is bounded above. I suspect this is true, and with a lot of hand waving can convince myself that the "ln" part of the RHS is "stronger" than the "^p" part and thus (ln(x+2))^p will eventually "slow down enough" so as to be less than x for large enough values of x.

But my log work leaves a little to be desired and I can't even prove there is a solution to the equation (ln(x+2))^p = x.

Any help much appreciated!


Justin
 
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  • #2
Never mind. :x
 
  • #3
There is a sweet test when it comes to series with logarithm in the argument. It says if a sequence a_n is decreasing and non-negative, then the series of a_n converges if and only if the series of 2^n*a_{2^n} does.

So here,

[tex]a_n=\frac{1}{\ln(n)^p}[/tex]

and so

[tex]2^na_{2^n} = \frac{2^n}{n^p\ln(2)^p}[/tex]

It's called the Cauchy condensation test: http://en.wikipedia.org/wiki/Cauchy_condensation_test
 
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  • #4
That IS sweet!

A gazillion thanks!

J
 

What is a sequence in lp spaces?

A sequence in lp spaces is a collection of numbers or functions that can be ordered according to some rule or pattern. In functional analysis, sequences are often studied in the context of lp spaces, which are a type of function space that consists of all measurable functions that satisfy a certain norm. In other words, a sequence in lp spaces is a sequence of functions that are defined on a given measure space and have a finite lp-norm.

What is the lp-norm?

The lp-norm is a mathematical measure of the size or magnitude of a vector or function in lp spaces. It is defined as the pth root of the sum of the absolute values raised to the pth power. In other words, for a sequence of numbers or functions (x1, x2, ..., xn), the lp-norm is given by ||x||p = (|x1|^p + |x2|^p + ... + |xn|^p)^(1/p). The lp-norm is used to measure the distance between two vectors or functions, and it plays a crucial role in the study of sequences in lp spaces.

What is the difference between l1 spaces and l2 spaces?

The main difference between l1 spaces and l2 spaces is the way the lp-norm is defined. In l1 spaces, also known as the space of absolutely summable sequences, the lp-norm is defined as the sum of the absolute values of the sequence. In l2 spaces, also known as the space of square-summable sequences, the lp-norm is defined as the square root of the sum of the squares of the sequence. This difference leads to different properties and applications of these spaces in functional analysis.

What are some common examples of sequences in lp spaces?

Some common examples of sequences in lp spaces include sequences of polynomials, trigonometric functions, and exponential functions. For instance, the sequence (1, 1/2, 1/3, ...) is a sequence in l1 space, while the sequence (1, 1/2, 1/4, ...) is a sequence in l2 space. These sequences are often used in the study of series and integrals in functional analysis.

What is the importance of studying sequences in lp spaces?

The study of sequences in lp spaces has many important applications in various areas of mathematics, including functional analysis, harmonic analysis, and probability theory. In particular, the lp spaces provide a natural framework for studying convergence and completeness of sequences of functions, which are essential concepts in many mathematical theories and applications. Furthermore, understanding sequences in lp spaces is crucial for the development of more advanced topics in functional analysis, such as Fourier analysis and operator theory.

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