Continuous Basis linear space

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1. Apr 12, 2017

Aroldo

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
$$\sum_{n} |\phi_{n}\rangle \langle \phi_{n}| = 1$$

3. The attempt at a solution

What is the dimension of this space?

By simple trigonometric relations, I found ${{1, (cos (x))^{2}, sin (x), cos (x), sin (2x)}}$ spam the space. Therefore, $dim = 5$.

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

Thank you

2. Apr 12, 2017

PeroK

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).

3. Apr 13, 2017

vela

Staff Emeritus
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.