TITLE: Inner product and norm of a function
QUESTION [5 upvotes]: 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?

REPLY [4 votes]: 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.