Is the Set {aX : a is in L_infinity} Necessarily Closed in L1 Spaces?

  • Thread starter DeanSkerl
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In summary, the conversation discusses the space of all d-dimensional bounded random vectors, L_infinity, and whether the set of all continuous left-multiplication orbits of a fixed d-dimensional random vector in L1, denoted as aX, is necessarily closed. The conclusion is that it is indeed closed.
  • #1
DeanSkerl
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Im an applied statistician who hasnt looked at this stuff in years (And even then never took a class in it formally). So try and forgive me if my language is sloppy.

Consider the space of all d-dimensional bounded random vectors, L_infinity. Its Banach, hence complete and closed.

Consider a single fixed d-dimensional random vector in L1. Let's call it X.

Is the set

{aX : a is in L_infinity} necessarily closed? Here I am considering the dot product so that aX is a random variable.

Any help, references, etc would be greatly appreciated.


--Scott
 
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  • #2
This is an orbit of ##X## for the continuous left-multiplication of ##L^\infty##, hence it is closed.
 

FAQ: Is the Set {aX : a is in L_infinity} Necessarily Closed in L1 Spaces?

What is an Lp space?

An Lp space is a mathematical concept used in functional analysis and measure theory to define a set of functions with certain properties. These spaces are named after the p-norm, which is a way of measuring the size or magnitude of a function.

How is an Lp space different from other function spaces?

An Lp space is different from other function spaces because it is defined by a specific p-norm, which takes into account the absolute value of a function raised to the power of p. This allows for a more precise and flexible definition of the space.

What is the significance of p in Lp spaces?

The value of p in Lp spaces determines the properties and characteristics of the functions within the space. For example, when p=2, the resulting space is called a Hilbert space, which has additional structure and properties compared to other Lp spaces.

What are some applications of Lp spaces?

Lp spaces have various applications in mathematics, physics, and engineering. They are commonly used in signal processing, image and audio compression, and probability theory. They also play a crucial role in the study of partial differential equations and harmonic analysis.

How are Lp spaces related to convergence?

Lp spaces are closely related to the concept of convergence. In particular, a sequence of functions converges in an Lp space if and only if it converges in the p-norm. This allows for the study of convergence in a more general and abstract setting, which has many applications in mathematics and other fields.

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