In spatial euclidean vector spaces norm is an intuitive concept: It measures the distance from the null vector and from other vectors.

The generalization to function spaces is quite a mental leap (at least for me).

**My question:**   
Is there some kind of intuition what "norm" or "euclidean norm" is even supposed to "mean" here? What is the null vector and how what does a "distance" between to functions reveal?

**Edit**: Please also see this follow up question: [Visualization of 2-dimensional function spaces](https://math.stackexchange.com/questions/19902/visualization-of-2-dimensional-function-spaces)