
Norms
Let $\displaystyle _{1}$ and $\displaystyle _{2}$ be the $\displaystyle L^1$norm and the $\displaystyle L^2$norm on $\displaystyle C[a,b]$ the space of continuous real valued functions on the closed interval $\displaystyle [a,b]$: explicitly
$\displaystyle f_{1}=\int_{a}^{b} f$, $\displaystyle f_{2}=\sqrt{\int_{a}^{b} f^2}$.
(a) Prove that $\displaystyle f_{1} \leq f_{2} $ for all $\displaystyle f \in C[a,b]$ (1). Hint: Remember the CauchySchwartz inequality for integrals.
(b) For $\displaystyle n \geq 1$, define $\displaystyle f_{n}:[0,1] \to \mathbb{R} $ by
$\displaystyle f_{n}(x)= \begin{matrix} n & if 0 \leq x \leq \frac{1}{n} \\ & \\ 2nn^{2}x & if \frac{1}{n} \leq x \leq \frac{2}{n} \\ & \\ 0 & if \frac{2}{n} \leq x \leq 1\end{matrix} $
Caculate $\displaystyle f_{n}_{1}$ and $\displaystyle f_{n}_{2}$.
So the thing is that I've been ill for over a week now and couldn't go to lectures resulting in me not having lecture notes and no notion of norms.
I have found hat CauchySchwartz states that $\displaystyle \int_{a}^{b} f(x)^{2} dx \int_{a}^{b} g(x)^{2} dx \geq (\int_{a}^{b}f(x) g(x) dx)^{2}$ for two real integrable functions in an interval [a,b]. I don't really know how to use this to prove (1).
As for (b) I am confused by the sequence of functions which I don't really know how to interpret (what is it geometrically?) or plug into $\displaystyle f_{1}=\int_{a}^{b} f$, $\displaystyle f_{2}=\sqrt{\int_{a}^{b} f^2}$.
Now, if someone could just start me off that would be great!

for the first one.
if $\displaystyle f\equiv0,$ it's a trivial case, so suppose $\displaystyle b>a,$ then $\displaystyle \int_{a}^{b}{\left f \right}=\int_{a}^{b}{\left f \right\cdot 1}\le \sqrt{\left( \int_{a}^{b}{\left f \right^{2}} \right)\left( \int_{a}^{b}{1^2} \right)},$ you can finish that now.

When I do that I end up with $\displaystyle \int_{a}^{b} f dx \leq \sqrt{ba} \sqrt{\int_{a}^{b} f^2 dx}$ but this does not necessarily imply $\displaystyle \int_{a}^{b} f dx \leq \sqrt{\int_{a}^{b} f^2 dx}$, does it?
I'm really tired so I might be missing something really obvious.