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In order to apply the ideas of vector spaces to functions, the text I have (Wavelets for Computer Graphics: Theory and Applications by Stollnitz, DeRose and Salesin) conveniently says

Since addition and scalar multiplication of functions are well defined, we can then think of each constant function over the interval $[0,1)$ as a vector, and we'll let $V^0$ denote the vector space of all such functions.

Ok, so I've heard this notion before, and it kind of makes sense. You want to apply the rules of vector spaces (and define things like inner product) for functions, so you go and say "a function is a vector."

But how does this mesh with the traditional physics definition of vector? A vector must have a magnitude and direction. What's going on here between algebra and physics?

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    I finally found something that made intuitive sense to me linking the traditionally taught idea of a vector to these vectors as functions. I could never quite understand where the integral product $\langle\cdot,\cdot\rangle=\int_0^1 f(x)g(x)dx$ came from. I was always told it was just defined that way, but the lecturers never explained why this might be the case... http://eng.fsu.edu/~dommelen/quantum/style_a/funcvec.html. Seems that a suitably well-behaved function defined over a finite interval $[0,1]$ can represent an infinite dimensional vector. Makes sense now.2015-10-26

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