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Math Help - Norms of a general vector space

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    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?
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    Re: Norms of a general vector space

    Quote Originally Posted by Euclid View Post
    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 L^2([a,b]) namely, \displaystyle \|f\|=\left(\int_a^b |f|^2\right)^{\frac{1}{2}} or more commonly the sup norm \|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.
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    Re: Norms of a general vector space

    One can show that the usual norms on a finite dimensional space, |v|= max |x_1|, |x_2|, ..., |x_n|, |v|= |x_1|+ |x_2|+ ...+ |x_n|, and \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, |v|_1= \sup_{x\in [a, b]} f(x), |v|_2= \int_a^b |f(x)|dx, and |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, v_1, would mark out the largest difference while the second, 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 |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 |v|_1.
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