Kunze Linear Algebra · Problem 3.5.4

Linear Transformations

Mathematics — Methods · dual-basis-integral-functionals

Problem

Let

V

be the vector space of all polynomial functions

p

from

R

into

R

which have degree 2 or less:

p(x) = c_0 + c_1 x + c_2 x^2.

Define three linear functionals on

V

f_1(p) = \int_0^1 p(x)\,dx, \quad f_2(p) = \int_0^2 p(x)\,dx, \quad f_3(p) = \int_0^{-1} p(x)\,dx.

Show that

\{f_1, f_2, f_3\}

is a basis for

V^*

by exhibiting the basis for

V

of which it is the dual.