check if a linear operator is bounded

Question

show that $Tf = f(0)$ is not a bounded linear functional on the space of continuous functions measured with the L2 norm, but it is a bounded linear functional if measured using the uniform norm.

Attempt: for the linear operator to be bounded, we need to show that $\frac{||Tf||}{||f||}\leq c <\infty$. but i really don't understand my space. the question says continuous function with L2 norm. but this is not a normed space since not all continuous functions are square integrable. so is my objects in this space square integrable continuous functions ?

Answer 1

For $C[0,1]$ you can put

$$f_n(x)=\begin{cases}-n^2x+n,&0\leq x\leq 1/n\\0,&1/n\leq x\leq 1\end{cases}$$

Then $\|f_n\|_2=\sqrt{n/3}$ and $|T(f_n)|/\|f_n\|_2=n/\sqrt{n/3}\to\infty$.

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On the other hand, $|T(f)|\leq \|f\|_{\infty}$.