If possible calculate the norms of the following linear forms. Let  be $I$ and $d$ defined as $$I:f\to\int_{-1}^{1} f(x)dx \\ d:f\to f(0)$$ on $\mathcal{C}^1[-1,1]$. Check if the linear forms are continous with respektive to the norms $\|\text{ }\|_{\infty}, \|\text{ }\|_{L_1}, \|\text{ }\|_{L_2}, \|\text{ }\|_{\mathcal{C}^1}$, and if so, calculate the norm.
I haven't quite understood norms in this way and so I have been searching all day on the internet for somekind of hint that could help me solve this problem. Sadly I wasn't lucky, so could someone here help me? Hints would be nice, so I can come to the solution myself.
 A: The question of continuity is one of "can we put the limit inside?" So what this question asks is:

"if $f_n\to f$ in the following norms, then is it true that $\displaystyle\int_{-1}^1f_n\to\displaystyle\int_{-1}^1 f$?"

So for the first one, if $f_n\to f$ uniformly, then is it true that $\displaystyle\int_{-1}^1f_n\to\int_{-1}^1f$?
Well, yes, the sequence is of uniformly continuous functions, so $|f_n-f|\le M$ for all $x\in(-1,1)$, hence LDCT implies
$$\left|\int_{-1}^1f_n-f\right|\le\int_{-1}^1|f_n-f|\to 0$$
For the norm, you want to find

$$\sup_{|f|\le 1}\left|\int_{-1}^1f\right|$$

but clearly $f\equiv 1$ gives the max, and a norm of $\displaystyle\int_{-1}^1 1=2$.
Similarly for the evaluation map we have $f_n(0)\to f(0)$ just by pointwise convergence, never mind the added bonus of uniform convergence. Similarly you want

$$\sup_{|f|\le 1}f(0)$$

and clearly $f(0)=1$ is as large as it can be, regardless of other points, so $d$ has norm $1$.
Can you see how this works for the others?
