All continuous functions on closed interval [a, b] form a vector space. The functions in this space are the vectors. However what is the physical significance of the norm of a vector in this space? For example if we found the norm of a function is 1/3 what does this signify? Does it dependent on the inner product used to define the norm?
Say if the inner product is the integral over [a, b] does this mean that the norm is related to the area?
One can show that the usual norms on a finite dimensional space, , , and are all "equivalent"- they give exactly the same limits, etc.
However, for infinite dimensional spaces, functions spaces in particular, the corresponding norms, , , and are very different.
Consider if you were trying to develop a mechanism to hold the temperature, T(t), in a room at some given function f(t). How would you measure the variation from the ideal? The first, , would mark out the largest difference while the second, , bases the "difference" on the average variation. If you were keeping, say, food, refrigerated, where a sudden, brief, spike in temperature won't hurt much but an extended lesser rise might, the first would be appropriate, then would be an appropriate measure. But if the problem were to keep an explosive cool, so that a sharp rise in temperature, no matter how short, would be disastrous, then we would measure using .