Dot Product Notation Clarification

1. Sep 14, 2009

CantorSet

The following notation is from the book "Frames and Bases."

Let f and g be vectors in $$R^{n}$$ with the usual dot product <,>.

Then, what does the notation $$\left|\left\langle f,g\right\rangle\right|^{2}$$ mean?

Specifically, does it mean $$\left|\sum^{n}_{i=1}f_{i} g_{i}\right|$$

or does it mean $$\left(\sum^{n}_{i=1}f_{i} g_{i}\right)^{2}$$

Last edited: Sep 14, 2009
2. Sep 14, 2009

lurflurf

so in such notation <f,g> is the dot product of f and g.
|<f,g>|^2 is the square of the absolute value of that dot product.
Some dot products are always real and the absolute value is redundent.
Other dot products can be complex and the absolute value is needed.

3. Sep 14, 2009

CantorSet

Thanks for responding. I was thinking that too. But it just seems inconsistent with the notation for the norm of a vector f in $$R^{n}$$.

For example, $$\left\langle \right f, f\rangle = \left\right\|f\|^{2}$$

In this case, f dot f without further squaring is equal to $$\left\right\|f\|^{2}$$.

But in |<f,g>|^2, you need compute f dot g, and THEN you still square its absolute value.

Is that correct?