Proving Completeness of Continuous Basis Vectors

In summary, the conversation discussed a vector space consisting of linear combinations of various trigonometric functions and its dimension. A possible set of basis vectors was suggested, but there was uncertainty about how to demonstrate completeness. The approach to proving completeness would likely involve explicitly constructing the identity operator and applying it to each basis vector.
  • #1
Aroldo
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Homework Statement


Consider the vector space that consists of all possible linear combinations of the following functions: $$1, sin (x), cos (x), (sin (x))^{2}, (cos x)^{2}, sin (2x), cos (2x)$$ What is the dimension of this space? Exhibit a possible set of basis vectors, and demonstrate that it is complete.

Homework Equations


$$\sum_{n} |\phi_{n}\rangle \langle \phi_{n}| = 1 $$

The Attempt at a Solution



What is the dimension of this space?[/B]
By simple trigonometric relations, I found [itex]{{1, (cos (x))^{2}, sin (x), cos (x), sin (2x)}}[/itex] spam the space. Therefore, [itex] dim = 5 [/itex].

I am not sure about how to proceed from here.
Aren't the basis [itex]{{1, (cos (x))^{2}, sin (x), cos (x), sin (2x)}}[/itex]?
How to demonstrate completeness of continuous bases?

Thank you
 
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  • #2
This is not a continuous basis. It is a finite basis. Showing it is "complete" is the same as showing it is a basis (unless you are using some odd terminology).
 
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  • #3
The intent is probably to have you explicitly construct the identity operator and apply it to each of the given functions and show that the operator is indeed the identity.
 

Related to Proving Completeness of Continuous Basis Vectors

1. What is a "Continuous Basis linear space"?

A Continuous Basis linear space is a mathematical concept used in linear algebra and functional analysis. It refers to a vector space where each vector can be expressed as a linear combination of an infinite number of basis vectors. These basis vectors form a continuous set, meaning there is no "gap" between them. This type of space is often used to model real-world phenomena that involve continuously varying quantities, such as time or space.

2. How is a Continuous Basis linear space different from a finite-dimensional vector space?

In a finite-dimensional vector space, the basis vectors are finite in number and can fully span the space. In a Continuous Basis linear space, the basis vectors are infinite and can only approximate the space. Additionally, in a finite-dimensional space, a vector can only be expressed as a linear combination of a finite number of basis vectors, while in a Continuous Basis space, a vector can be expressed as a linear combination of an infinite number of basis vectors.

3. What are some applications of Continuous Basis linear spaces?

Continuous Basis linear spaces are often used in physics and engineering to model physical systems and phenomena that involve continuously varying quantities. Examples include the wave functions used in quantum mechanics, the Fourier transform used in signal processing, and the heat equation used in thermodynamics.

4. Can a Continuous Basis linear space have a finite basis?

No, a Continuous Basis linear space cannot have a finite basis. This is because the basis vectors must be continuous, meaning there are no gaps between them. If the basis were finite, there would be gaps between the basis vectors, and the space would not be continuous.

5. How are Continuous Basis linear spaces related to Hilbert spaces?

Hilbert spaces are a type of Continuous Basis linear space that have additional properties, such as being complete and having an inner product defined on them. Not all Continuous Basis spaces are Hilbert spaces, but all Hilbert spaces are Continuous Basis spaces. Hilbert spaces are often used in functional analysis and quantum mechanics.

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