Absolutely Continuous Functions

In summary, the completeness of the space of all absolutely continuous functions depends on the chosen norm. The induced norm from a specific inner product, as mentioned in the conversation, may not be suitable for showing completeness. Instead, a different norm must be chosen and proven to be complete through the convergence of any absolutely convergent series. This can be done by showing that a Cauchy sequence in the space converges. A link to a proof is provided for further reference.
  • #1
Matthollyw00d
92
0
Is the space of all absolutely continuous functions complete?
I've never learned about absolutely continuous functions, and so I'm unsure of their properties when working with them. I'm fairly certain it is, but would like some verification.
Or a link to something on them besides the wikipedia page could be useful as well.

Much appreciated.
 
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  • #2
It depends on what you use for the norm. For example (for simplicity I'll assume a finite domain) - if ||f||=max|f|, the answer would be no. On the other hand if ||f||=max|f|+max|f'|, the answer would be yes.
 
  • #3
I was thinking with the induced norm from [tex]<f,g>=\Sigma_{k=1}^n \int^1_0 f^{(k)}(t)\overline{g^{(k)}(t)}dt[/tex]
 
  • #4
I'm confused? How are f(k) related to f?
 
  • #5
The kth derivatives.
I'm dealing with a subset of absolute continuous functions and trying to show completeness. It seemed easier to just show it was closed if I knew the space of all absolutely continuous functions were complete w.r.t the induced metric from the above inner product. But now I don't think we can talk about that space w.r.t the inner product, since absolute continuity gives nothing about the kth derivatives existing.

I guess I'm left with showing that an arbitrary Cauchy sequence converges.
 
  • #6
What is n? And indeed, absolutely continuous functions need not have k-th derivatives (for any k), so this inner product does not make any sense.

So back to start: what norm do you want to use (you can't talk about completeness without talking about a norm)?
 
  • #7
Well I was dealing with a subset where it did make sense.

Let me just ask a different question:
If [tex]\{h_n\}[/tex] is a sequence in H and [tex]\Sigma^\infty_{n=1}||h_n||<\infty[/tex] and I show that [tex]\Sigma^\infty_{n=1}h_n[/tex] converges in H; does that imply that H is complete?
 
  • #8
Assuming that (h_n) is an arbitrary sequence: yes, one of the equivalent formulations of completeness of a normed space is "every absolutely convergent series converges".

\\edit: for a proof of this, see e.g. here.
 
Last edited:

1. What is an absolutely continuous function?

An absolutely continuous function is a type of function that is defined on a closed and bounded interval and has the property that for any given interval within that interval, the function's variation can be made arbitrarily small by choosing a small enough subinterval. In simpler terms, this means that the function is smooth and does not have any sudden changes or jumps.

2. How is an absolutely continuous function different from a continuous function?

An absolutely continuous function is a type of continuous function, but it has the additional property of being able to be broken down into smaller and smaller subintervals with small variations. This is not necessarily true for all continuous functions, as some may have sudden jumps or discontinuities within their intervals.

3. What is the significance of absolutely continuous functions in mathematics?

Absolutely continuous functions are important in mathematics because they have many useful properties that make them easier to work with and allow for more precise analysis. They are also closely related to other important concepts such as derivatives, integrals, and measure theory.

4. Can all functions be classified as absolutely continuous?

No, not all functions can be classified as absolutely continuous. In order for a function to be absolutely continuous, it must meet certain criteria and have specific properties. Functions that have sudden jumps or discontinuities, or have large variations within their intervals, cannot be considered absolutely continuous.

5. How are absolutely continuous functions used in real-world applications?

Absolutely continuous functions are used in many real-world applications, particularly in areas such as physics, engineering, and economics. They are helpful in modeling and predicting continuous processes, such as changes in temperature or stock prices, and can also be used to optimize and solve problems in these fields.

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