Let f : [0, 1] → $\mathbb{R}$ be continuous. Then we can define the function $T : C[0, 1]$ → $\mathbb{R}$ by:
$T(f)$ = $\int_\limits{0}^{1}(1+x^2)f(x)dx$ (since the product of continuous functions is continuous).
Show that there exists a constant K with the property that for all $f\in C[0, 1]$, $|T(f)|$ ≤ $K||f||_\infty$ $\leftarrow$ (refers to sup norm).
Use the result to prove that if {$f_n$}$_{n=1}^{\infty}$ is a sequence in C[0, 1] such that $f_n$ → f in the sup norm, then the sequence of numbers $T(f_n)$ converges to the number $T(f)$.
I am not really sure how to go about starting this problem.
$$|T(f)|=|\int_0^1(1+x^2)f(x)dx|$$
$$\leq ||f||_\infty \int_0^1(1+x^2)dx$$
and you take $K=\int_0^1(1+x^2)dx=\frac{4}{3}$.