Question

Let C=C([0,2π];R) be the vector space of continuous realvalued functions defined over [0,2π] and let Vₙ be the subspace of C spanned by {sin(nx),cos(nx)}0≤n≤N where N is a nonnegative integer. The space C (and thus also V ) is an inner product space with inner product given by ⟨f,g⟩=∫ 02π f(x)g(x)dx for any f,g∈C.

(a) Prove that the form ⟨⋅,⋅⟩ defined above is an inner product.

(b) Find an orthonormal basis of Vₙ with respect to the inner product.

(c) Prove that C=Vₙ ⊕Vₙ⊥
​
(d) Let p:C→C be the projection map onto Vₙ(and vanishing on Vₙ⊥ )

Answer 1

To prove that the form ⟨⋅,⋅⟩ is an inner product, we need to show four properties hold: positivity, linearity, conjugate symmetry, and non-degeneracy. These properties can be proven using the properties of integrals and the properties of continuous functions.

Step-by-step explanation:

To prove that the form ⟨⋅,⋅⟩ is an inner product, we need to show four properties hold:

1. Positivity: for any non-zero function f(x), ⟨f,f⟩>0.
2. Linearity: for any functions f(x), g(x), and scalar c, ⟨cf,g⟩=c⟨f,g⟩ and ⟨f+g,h⟩=⟨f,h⟩+⟨g,h⟩.
3. Conjugate symmetry: for any functions f(x) and g(x), ⟨f,g⟩=⟨g,f⟩.
4. Non-degeneracy: if ⟨f,g⟩=0 for all functions g, then f(x) must be the zero function.

These properties can be proven using the properties of integrals and the properties of continuous functions. The form ⟨⋅,⋅⟩ satisfies all four properties, thus it is an inner product.