# Thread: Norms of a general vector space

1. ## Norms of a general vector space

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?

2. ## Re: Norms of a general vector space

Originally Posted by Euclid
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! Of course it depends how you define the norm! There are several nice ways to define a norm on this space! For example, there is the norm it inherits as being a subspace of $\displaystyle L^2([a,b])$ namely, $\displaystyle \displaystyle \|f\|=\left(\int_a^b |f|^2\right)^{\frac{1}{2}}$ or more commonly the sup norm $\displaystyle \|f\|_\infty=\sup |f(x)|$. If by significance you mean what it 'means' if two elements are close in norm? Well, why don't you tell us.

3. ## Re: Norms of a general vector space

One can show that the usual norms on a finite dimensional space, $\displaystyle |v|= max |x_1|, |x_2|, ..., |x_n|$, $\displaystyle |v|= |x_1|+ |x_2|+ ...+ |x_n|$, and $\displaystyle \sqrt{x_1^2+ x_2^2+ ...+ x_n^2}$ are all "equivalent"- they give exactly the same limits, etc.

However, for infinite dimensional spaces, functions spaces in particular, the corresponding norms, $\displaystyle |v|_1= \sup_{x\in [a, b]} f(x)$, $\displaystyle |v|_2= \int_a^b |f(x)|dx$, and $\displaystyle |v|_3= \sqrt{\int_a^b f^2(x)dx}$ 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, $\displaystyle v_1$, would mark out the largest difference while the second, $\displaystyle v_2$, 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 $\displaystyle |v|_2$ 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 $\displaystyle |v|_1$.