For norms on $C[a,b]$
I'm having trouble picturing how the second norm looks like
If the first norm $||f||_1 = \int _{a}^b |f(t)|dt $ is represented by the shaded region
and the infinite norm $||f||_\infty = max_{t \in [0,1]} |f(t)| $ is represented by the max distance from t axis to f
what would the second norm $ ||f||_2 = \sqrt{\int _{a}^b f(t)^2dt} $ look like?
If you forget for a moment the square root, you can visualize $f(x)^2$ as as the area of the square of side length $f(x)$. Thus your are actually summing up all these areas, hence you can consider it as if it were a volume, see picture