What Is the Physical Significance of the Norm of a Function in a Vector Space?

[matqkks]

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?

 

[micromass]

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?

Yes, if you define the inner product as

$$<f,g>=\int_a^b{f(t)g(t)dt}$$

then your norm will be

$$\|f\|_2=\sqrt{\int_a^b{|f(t)|^2dt}}$$

This means that a function f will be close to 0 if the area of f is very low. More generally, a function f will be close to g if their area's are close together.

You have different possible norms on the continuous functions, and all describe something different. Good questions you should ask for each norm is "what functions are close to the 0 function" or "when are two functions close together".

 

[mathwonk]

norms measure the size of things. integral norms for functions measure the average value, max norms measure the maximum value, integrals of squares measure the average squared value. you must decide in a physical situation which of these measures suits your problem.

 

[HallsofIvy]

If you have an inner product, then there is a standard way of defining the norm and so the "length" of a vector. However, it is possible to have a norm without an inner product.

$L_1([a, b])$ is the set of functions, f(x), such that the Lebesque integral, $\int |f(x)|dx$ exists. And, of course, we define the norm of f to be that integral.

The crucial point of the norm of a function is that it allows us to measure the distance between functions, allowing us to talk about convergence of sequences of functions.

 

[mathwonk]

as halls says, inner products are more special than norms, and allow also to measure angles.