Reflexivity of L^p and its Implications for Integration

  • Thread starter Goklayeh
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In summary, L^p space is a vector space of measurable functions with finite p-norms, used to study functions and measure their properties. It has applications in various fields such as integral and differential equations, probability and statistics, and is characterized by its p-norm which sets it apart from other function spaces. L^p space is also a Banach space, allowing for the development of powerful mathematical tools.
  • #1
Goklayeh
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Could someone confirm or refute the following statement?

[tex]f \in L^p\left(X, \mu\right) \: \Leftrightarrow \: \int_X{\lvert fg \rvert d\mu < \infty\: \forall g \in L^q\left(X, \mu\right)[/tex]

where [tex]1<p<\infty,\: \frac{1}{p}+\frac{1}{q}=1[/tex] and [tex](X, \mu)[/tex] is a measurable space (of course, the [tex](\Rightarrow)[/tex] is trivial by Holder inequality)

Thanks in advance!
 
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  • #2
It looks correct to me. From my recollection. Lp and Lq are adjoint, when p, q > 1 and 1/p + 1/q = 1.
 
  • #3
For those values of p, Lp is reflexive. What can you infer from this?
 

1. What is L^p space?

L^p space is a mathematical concept used in functional analysis and measure theory. It is a vector space of all measurable functions f for which the p-norm of f is finite. The p-norm is calculated by raising the absolute value of each point in the function to the power of p, adding all the values, and then taking the p-th root.

2. What is the significance of L^p space?

L^p space is used to study the properties of functions and measure their sizes and behaviors. It is particularly useful in the study of integral and differential equations, as well as in probability and statistics.

3. How is L^p space different from other function spaces?

L^p space is different from other function spaces because it is characterized by the p-norm, which allows for the comparison and measurement of functions based on their sizes. Other function spaces, such as L^infinity and L^1, have different characteristics and are used for different purposes.

4. What are the applications of L^p space?

L^p space has a wide range of applications in mathematics, physics, engineering, and other fields. It is used in the study of Fourier series, signal processing, and control theory. It also has applications in probability and statistics, such as in the study of probability distributions and random processes.

5. Is L^p space a Banach space?

Yes, L^p space is a Banach space, which means it is a complete normed vector space. This means that every Cauchy sequence in L^p space converges to a limit that is also in L^p space. This property is important in the study of functional analysis and allows for the development of powerful mathematical tools and techniques.

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