Riesz representation theorem application

Question

I need help with the second question below

My thoughts: Taking the first functional, by the riesz representation theorem, I can find a signed measure $v$ on $B[0,1]$ such that $l_1(p)=$ the integral of $c_0$ w.r.t $v$ between $0$ and $1$. But I don't know what to do next.

Thanks

Answer 1

Along the lines of Nate's hint: consider the Dirac Measure $$\delta_x(A) = 1_A(x) = \begin{cases} 1 & x \in A\\ 0 & \text{otherwise} \end{cases} $$ First, do so with $x = 0$, then with $x = 1$.

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For (i): we define the measure $$\delta_0(A) = 1_A(0) = \begin{cases} 1 & 0 \in A\\ 0 & \text{otherwise} \end{cases} $$ We find that for polynomials $p(x) \in P$: $$\int_{[0,1]} p(x)\,d\delta_0 = p(0) = c_0 = l_1(p)$$ as desired. We may extend this functional to other continuous functions by applying the same integral. That is, for $f \in E$, we define $$l_1(f) = \int_{[0,1]}f(x)\,d\delta_0 = f(0)$$