Proving Completeness of Continuous Basis Vectors

[Aroldo]

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 ${{1, (cos (x))^{2}, sin (x), cos (x), sin (2x)}}$ 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

 

[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).

 

[vela]

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.