Inner product and norm of a function

I have recently started a undergraduate 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:

\begin{align} \langle f, g \rangle &= \int_a^b f(t) g(t) \,\mathrm{d}t \\ \| f \| &= \sqrt{\langle f, f \rangle} = \sqrt{\int_a^b f^2(t) \,\mathrm{d}t} \end{align}

Concerns:

What does it mean if $\int_a^b f^2(t) \,\mathrm{d}t < 0$?

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?

1 Answer

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.