Hilbert Space: f(x) = x^n on Interval (0,1)

In summary, a Hilbert Space is a complete vector space with an inner product, named after mathematician David Hilbert. It is used in functional analysis and quantum mechanics. The function f(x) = x^n on the interval (0,1) can be represented as an element in a Hilbert Space and can be used to construct a basis for the space. The Hilbert Space is related to other mathematical concepts such as Banach Spaces, and has applications in physics, engineering, and machine learning.
  • #1
Gumbercules
11
0

Homework Statement


for what range of n is the function f(x) = x^n in Hilbert space, on the interval (0,1)? assume n is real.


Homework Equations



functions in Hilbert space are square integrable from -inf to inf

The Attempt at a Solution


I am having trouble with the language of the problem and the concept of the Hilbert space in general. Does 'on the interval (0,1)' mean that the square integrable function only has to converge for limits of integration between 0 and 1?
 
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  • #2
Yes they are asking you for what n the following integral converges.

[tex]
\int_0^1 |f(x)|^2 dx <\infty
[/tex]
 
  • #3
Thanks!
 

FAQ: Hilbert Space: f(x) = x^n on Interval (0,1)

What is a Hilbert Space?

A Hilbert Space is a mathematical concept that describes a complete vector space with an inner product. It is named after the German mathematician David Hilbert and is used in many areas of mathematics, including functional analysis and quantum mechanics.

What does the function f(x) = x^n on the interval (0,1) have to do with Hilbert Spaces?

The function f(x) = x^n on the interval (0,1) is an example of a function that can be represented as an element in a Hilbert Space. This function, along with other functions, can be used to construct a basis for the Hilbert Space. This allows us to represent other functions in the Hilbert Space as a linear combination of these basis functions.

How is the Hilbert Space related to other mathematical concepts?

The Hilbert Space is closely related to other mathematical concepts such as Banach Spaces, which are complete normed vector spaces. In fact, every Hilbert Space is also a Banach Space, but the converse is not always true.

What is the significance of the interval (0,1) in the function f(x) = x^n on the interval (0,1)?

The interval (0,1) in the function f(x) = x^n on the interval (0,1) represents the domain of the function. This means that the function is only defined for values of x between 0 and 1. This interval is often used in mathematical analysis because it is a convenient range for many functions.

How is the concept of Hilbert Spaces used in real-world applications?

Hilbert Spaces are used in many real-world applications, particularly in physics and engineering. For example, in quantum mechanics, Hilbert Spaces are used to model the states of a quantum system. In signal processing, Hilbert Spaces are used to analyze and process signals. In machine learning, they are used to represent data and make predictions. In summary, Hilbert Spaces have a wide range of applications in various fields of science and technology.

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