# 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?

• $\int_{a}^{b}f^2(t)dt$ is always positive for $a<b$. Apr 12, 2013 at 18:43
• 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? Apr 12, 2013 at 18:49
• The standard interval notation $[a,b]$ means $a<b$. Apr 12, 2013 at 18:50
• @AdamYac : there are many inner products on $V$. Whoever wrote that definition was sloppy. Apr 12, 2013 at 22:46

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.