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Homework Help: Continuous Basis linear space

  1. Apr 12, 2017 #1
    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 [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
  2. jcsd
  3. Apr 12, 2017 #2


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    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).
  4. Apr 13, 2017 #3


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    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.
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