TITLE: Intuitive explanation of $L^2$-norm
QUESTION [8 upvotes]: I have to play a lot with the $L^2$-norm defined as $\|w\|=\sqrt{\int_a^b \langle f,f \rangle}$. However, I don't understand the interpretation of that norm.
We know that the Euclidean norm measure the length of a vector in the Euclidean space, but what does the $L^2$-norm? 
Is there anyone could give me an intuitive (even a graph, if possible) explanation of that norm?

REPLY [9 votes]: There are several good answers here, one accepted. Nevertheless I'm surprised not to see the $L^2$ norm described as the infinite dimensional analogue of Euclidean distance. 
In the plane, the length of the vector $(x,y)$ - that is, the distance between $(x,y)$ and the origin - is $\sqrt{x^2 + y^2}$. In $n$-space it's the square root of the sum of the squares of the components.
Now think of a function as a vector with infinitely many components (its value at each point in the domain) and replace summation by integration to get the $L^2$ norm of a function. 
Finally, tack on the end of last sentence of @levap 's answer:

... the $L^2$ norm has the advantage that it comes from an inner
  product and so all the techniques from inner product spaces
  (orthogonal projections, etc) can be applied when we use the $L^2$
  norm.