Dual Spaces

  1. Consider the vector space V = ℝ≤2[x] of at most quadratic polynomials over ℝ on the interval [−1, 1] and define the maps φk : V ⇾ ℝ by \[ \varphi_k (p) = \langle \varphi \mid p \rangle = \int_{-1}^1 {\text d}x\, x^k p(x), \qquad k=0,1,2,\ldots . \]
    1. Show that the φk are linear functionals on V.
    2. Show that {φ₀, φ₁, φ₂} is a basis of V*.
    3. Find the basis of V* dual to the monomial basis {1, x, x²}.