Question

Which of the following maps on P(R) are linear functionals? (a) p(x)↦∫ˣ ₋₁​ p(t)dt (b) p(x)↦∫² ₀ t²p(2t ³−1)dt (c) p(x)↦p ′′ (π) (d) p(x)↦2p(1)x ²

Answer 1

The maps on P(R) that are linear functionals are (a) p(x)↦∫ˣ ₋₁​ p(t)dt, (c) p(x)↦p′′(π), and (d) p(x)↦2p(1)x².

Step-by-step explanation:

A linear functional is a linear transformation from a vector space to the field of scalars. In this case, the vector space is P(R), the space of all polynomials with real coefficients, and the field of scalars is also R. To determine which maps on P(R) are linear functionals, we need to check if they satisfy the properties of linearity.

(a) p(x)↦∫ˣ ₋₁​ p(t)dt: This map is a linear functional because it satisfies the properties of linearity, namely, additive and homogeneous.

(b) p(x)↦∫² ₀ t²p(2t³−1)dt: This map is not a linear functional because it fails to satisfy the property of homogeneity.

(c) p(x)↦p′′(π): This map is a linear functional because it satisfies the properties of linearity.

(d) p(x)↦2p(1)x²: This map is a linear functional because it satisfies the properties of linearity.