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I have recently started a undergrad. linear algebra course in which these definitions came up:

Let $V$ be the vector space $C[a,b]$ of all continuous functions on $[a,b]$. Then the inner product and norm are defined as:

$\left \langle f,g \right \rangle=\int_{a}^{b}f(t)g(t)dt$

$\left \| f \right \|=\sqrt{\left \langle f,f \right \rangle}=\sqrt{\int_{a}^{b}f^{2}tdt}$

Concerns:

What does it mean if $\int_{a}^{b}f^2(t)dt<0?$

It is also,strangely,possible to calculate the angle between functions(non-linear), is this considered the average angle in $[a,b]$ or what is it's geometrical representation?

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  • $\begingroup$ $\int_{a}^{b}f^2(t)dt$ is always positive for $a<b$. $\endgroup$ – Learner Apr 12 '13 at 18:43
  • $\begingroup$ I did not know that. But is not $C[1,0]$ the same vector space? Or is it a necessity that a<b when vector spaces are defined? $\endgroup$ – EricAm Apr 12 '13 at 18:49
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    $\begingroup$ The standard interval notation $[a,b]$ means $a<b$. $\endgroup$ – Learner Apr 12 '13 at 18:50
  • $\begingroup$ @AdamYac : there are many inner products on $V$. Whoever wrote that definition was sloppy. $\endgroup$ – Stefan Smith Apr 12 '13 at 22:46
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It is important to note that the (inner-product) space you are working with is not the product of $[a,b]$ and the image of functions $f$. It is the infinite dimensional space of continuous functions defined on $[a,b]$. You should picture each $f$ in that space as an infinite vector.

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  • $\begingroup$ Alright, I did not have that picture in mind but I do see your point. A question I should not ask but since we can deal with all kinds of C^0-functions; I know a vector have magnitude and direction but is it necessary a straight line? $\endgroup$ – EricAm Apr 12 '13 at 18:56
  • $\begingroup$ Or should I imagine all kinds of elementary functions as infinite straight vectors? In that case the angle makes more sense.. $\endgroup$ – EricAm Apr 12 '13 at 18:59
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    $\begingroup$ Yes, linearity here comes the view that you treat each function as an infinite vector. The area of mathematics that studies this (among other things) is "functional analysis". $\endgroup$ – Learner Apr 12 '13 at 19:03

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