Proving Completeness of a function space

In summary, the conversation discusses the completeness of a set F under a given norm. The approach of looking at the pointwise limit of an arbitrary Cauchy sequence is mentioned, but it is not proven that it converges in the metric induced by the norm. An outline of a potential proof is given involving a Cauchy sequence (f_{n}) and a function g_{n}(x) = f_{n}(x)/(x^{2} + 1), and it is concluded that f \in F. The conversation ends with a thank you and a Merry Christmas greeting.
  • #1
vineethbs
8
0
Let [tex]F = \left\{f : [0, \infty) \rightarrow R, norm(f) = \sup_{x \in [0,\infty)} \frac{|f(x)|}{x^{2} + 1} < \infty\right\}[/tex]

Is F complete , under the given norm ?

My approach was to look at the pointwise limit of an arbitrary Cauchy sequence, but I am not able to prove that it converges in the metric induced by the norm.

Thank you and Merry Christmas !
 
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  • #2
Hi , is the following correct ?
(an outline of the proof )
Given an arbitrary Cauchy sequence (f_{n})
we have that
[tex] \forall \epsilon > 0, \exists n_{\epsilon} \leq m < n \, s.t \sup_{0 \leq x < \infty} \frac{|f_{n}(x) - f_{m}(x)|}{x^{2} + 1} < \epsilon [/tex]

[tex] g_{n}(x) = f_{n}(x)/(x^{2} + 1) [/tex]

this means that [tex] \sup_{ 0 \leq x \infy} |g_{n}(x) - g_{m}(x)| < \epsilon [/tex] for m, n as above

so that [tex] g_{n}(x) \rightarrow g(x) , \forall x, uniformly [/tex]
which means that [tex] \sup_{0 \leq x < \infty} |g_{n}(x) - g(x)| \rightarrow 0 [/tex]

with [tex] f(x) = g(x)(x^{2} + 1) [/tex]

[tex] \sup_{0 \leq x < \infty} |\frac{f_{n}(x) - f(x)}{x^{2} + 1}| \rightarrow 0 [/tex]
[tex] \forall x, \frac{|f(x)|}{x^{2} + 1} \leq \frac{|f_{n}(x) - f(x)|}{x^{2} + 1} + \frac{|f_{n}(x)|}{x^{2} + 1} [/tex] which should give that f \in F ?
Thank you for your time !
 
Last edited:

Related to Proving Completeness of a function space

1. What is the definition of completeness in a function space?

Completeness in a function space means that all possible functions in that space can be expressed as a linear combination of a set of basis functions. This implies that the function space is able to represent all functions within its domain.

2. Why is proving completeness important in mathematics?

Proving completeness is important in mathematics because it allows us to determine the extent to which a function space can represent all possible functions in a given domain. This is useful in various fields such as physics, engineering, and computer science, where accurate mathematical representations of real-world phenomena are necessary.

3. What is the process for proving completeness of a function space?

The process for proving completeness of a function space involves two steps: first, showing that the set of basis functions span the entire function space, meaning that any function in the space can be expressed as a linear combination of the basis functions. Second, showing that the set of basis functions is linearly independent, meaning that no basis function can be expressed as a linear combination of the others.

4. Can a function space be complete in one domain but not in another?

Yes, a function space can be complete in one domain but not in another. This depends on the set of basis functions chosen and the properties of the domain. For example, a set of basis functions may be complete in the space of continuous functions, but not in the space of differentiable functions.

5. Are there different ways to prove completeness of a function space?

Yes, there are different methods for proving completeness of a function space. Some of the common techniques include using the Gram-Schmidt process, using the Stone-Weierstrass theorem, or using the Riesz representation theorem. The method used may depend on the specific function space and its properties.

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