Calculating angles - what's my mistake? Let $C[a,b]$ be the space of all continuous functions $[a,b]\rightarrow \mathbb R$. Then, functional
$$||f||=\int^b_a|f(x)| dx$$
Clearly satisfies all the axioms of a norm, with its corresponding dot product given by: $\left<f,g\right>=\frac12(||f+g||^2-||f||^2-||g||^2)$
Then, $\frac{\left<f,g\right>}{||f||\cdot||g||}$ should give the angle between $f$ ang $g$. But, if $f,g>0$, we have:
$\left<f,g\right>\overset{def}=\frac12((\int^b_af(x)+g(x) dx)^2-(\int_a^b f(x) dx)^2-(\int_a^bg(x) dx)^2)=\int_a^bf(x)dx\int_a^bg(x)dx$
$||f||\cdot||g||\overset{def}=\int_a^b f(x) dx \int_a^bg(x) dx$
$\frac{\left<f,g\right>}{||f||\cdot||g||}=1$
So it would mean that all continouous functions with positive values are parallel to each other in this norm, which is nonsense.
Where's my mistake? How can I properly calculate angles in $(C[a,b],||\cdot||)$? I know that the error must be something trivial, but I cannot find it.
Edit: Let $v,w \in V: ||v||=||w||$. I know that element of $\text{span}(v)$ closest to $w$ is in the form $\theta v$, $\theta \in [0,1]$. If dot product is not a legit option, what is a general way of finding $\theta$? 
 A: You dropped absolute values in your computation of $\langle f,g\rangle$ and $||f||\cdot||g||$, but when you put them back in, the dot product is not bilinear, which is a problem. And if you do leave them out, then it is not positive-definite:
$$||f|| = \sqrt{\langle f,f \rangle} = \sqrt{\left(\int_a^b f(x) \, dx\right)^2} = 0$$
does not imply $f=0$.
You might want
$$||f|| := \int_a^b f(x)^2 \, dx$$
and
$$\langle f,g \rangle := \int_a^b f(x)g(x) \, dx.$$
A: The scalar product you introduce is not a scalar product.
A: Not every norm is compatible with an inner product, and you have discovered an example.
Here's a simpler but very closely related example. Define the sup norm on $\mathbb R^2$:
$$\|(x,y)\| = |x| + |y|
$$
Suppose one adopts the same formula for the inner product of two vectors as given in your question. If you then take two positive vectors $\vec v_1 = (x_1,y_1)$ and $\vec v_2 = (x_2,y_2)$, you will similarly discover that the quantity $\frac{\langle \vec v_1,\vec v_2 \rangle}{\|v_1\| \|v_2\|}$ misbehaves.
You can see what is happening on a geometric level by examining the unit ball of the norm $\|(x,y) \| = |x| + |y|$: it is a diamond, not an ellipse as it would be if it were defined by a positive definite inner product. On an algebraic level, the explanation is that your inner product is not bilinear, as is said by @csprun. 
