$C_0([0,\pi])=\{f:[0,\pi] \to \mathbb{R} \mid f$ is continuous and $ f(0)=0=f(\pi) \} $
For $f \in C_0([0,\pi])$, define Fourier parcial sums of $f$ by:
$$ S_N(f)= \sqrt\frac{2}{\pi}\sum_{k\geq1} b_k f(x) \sin(kx) $$ where the coefficient of Fourier of $f$ is $b_k= \displaystyle \sqrt\frac{2}{\pi} \int_0^\pi f(x) \sin(kx)dx$.
Define:
$$ \|f\|_{H^1}=\left(\sum_{k\geq1}(1+k^2)b_k^2\right)^{1/2} $$
and
$H_0^1=\{f \in C_0([o,\pi]): \|f\|_{H^1}<+\infty\}$.
Proof that $\|*\|_{H^1}$ defines a norm.
Well I want to prove this without having to consider if $f$ is differentiable or not, since in the case it is, $\|f\|_{H^1}=\displaystyle \int_0^\pi \left( f'(x)^2 + f(x)^2\right) dx$.
We have, for $f,g \in H_0^1$, with coefficients of Fourier $b_k$ and $c_k$, respectivaly, and $\lambda \in \mathbb{R}$:
$1) \|f\|_{H^1}$ is well-defined since $(1+k^2)b_k^2 \geq 0 \ \forall_{k \in \mathbb{N_1}}$;
$2)\|f\|_{H^1} \geq0$ and $\|f\|_{H^1}=0 <=> (1+k^2)b_k^2=0 \ \forall_{k \in \mathbb{N_1}}<=> b_k=0 \ \forall_{k \in \mathbb{N_1}} <=> f(x) \sin(kx)=0 \ \forall_{k \in \mathbb{N_1}} <=> f \equiv 0$;
$3) \|\lambda f\|_{H^1}^2=\displaystyle \sum_{k \geq1}(1+k^2)(\lambda b_k)^2= \lambda^2 \|f\|_{H^1}^2$, the first $=$ because $\displaystyle \int_0^\pi \left(\lambda f(x) \right) \sin(kx)dx=\lambda \displaystyle \int_0^\pi f(x) \sin(kx)dx$, so $\|\lambda f\|_{H^1}= |\lambda| \|f\|_{H^1}$;
$4) \|f+g\|_{H^1}^2=\displaystyle \sum_{k \geq1}(1+k^2)(b_k+c_k)^2=\displaystyle \sum_{k \geq1}(1+k^2)b_k^2+\sum_{k \geq1}(1+k^2)c_k^2+2\sum_{k \geq1}(1+k^2)b_k c_k$
The first two sums are $\|f\|_{H^1}^2$ and $\|g\|_{H^1}^2$.
I'm told to use the Cauchy-Schwarz inequality for series: if $(u_k)_k,(v_k)_k$ are real sequences, then
$$ \left( \sum_k |u_k||v_k| \right)^2 \leq \left( \sum_k u_k^2 \right) \left( \sum_k v_k^2 \right) $$
Using $u_k=(1+k^2)^{1/2}b_k, \ v_k=(1+k^2)^{1/2}c_k$, we get:
$$ 2\sum_{k \geq1}(1+k^2)b_k c_k \leq 2 \left( \sum_{k\geq1}(1+k^2)b_k^2 \right)^{1/2} \left( \sum_{k\geq1}(1+k^2)c_k^2 \right)^{1/2}. $$
The second member is equal to $2 \|f\|_{H^1} \|g\|_{H^1}$.
So,
$$ \|f+g\|_{H^1}^2 \leq \|f\|_{H^1}^2 + \|g\|_{H^1}^2 + 2 \|f\|_{H^1} \|g\|_{H^1} $$
this is
$$ \|f+g\|_{H^1}^2 \leq \left( \|f\|_{H^1} + \|g\|_{H^1} \right)^2. $$
Since $\|*\|_{H^1} \geq0$, we have $\|f+g\|_{H^1} \leq \|f\|_{H^1} + \|g\|_{H^1}$.
This concludes that $\|*\|_{H^1}$ is a norm on $H_0^1$. Is this correct? I was having trouble with the last one, until I passed from the sum to the norm...