Just as we have orthogonal vectors/vector spaces/etc., we can have orthogonal functions/function spaces/etc. I'm trying to apply these concepts to physical processes. Here's a general idea of what I'm doing:(adsbygoogle = window.adsbygoogle || []).push({});

Suppose you have a physical quantity you are trying to measure, ##F##, and it depends on two other physical processes, ##A## and ##B##.

Consider the case where ##A## and ##B## are completely independent from one another:

We apply process ##A## only and measure ##F_A##.

Next, we apply process ##B## and measure ##F_B##.

We then apply processes ##A## and ##B## simultaneously and measure ##F_{AB}##.

Since the processes are independent, we expect to find that ##F_A + F_B = F_{AB}##.

We could say that processes ##A## and ##B## are "orthogonal", and that their inner product is

$$\langle A \cdot B \rangle = 0$$

Contrast this to the case where ##A## and ##B## are not completely independent:

$$ F_A + F_B \neq F_{AB}$$

and their inner product is

$$\langle A \cdot B \rangle \neq 0$$

Is it accurate to say that the inner product is a quantity that represents "dependency" between processes ##A## and ##B##? Or is there a more specific physical interpretation? And, can the value of the inner product be used to say anything in particular about the processes ##A## and ##B##?

When dealing with vectors, the inner product gives information about the angle between the two vectors, which has physical significance for the vectors. When we apply concepts of "dependency" and "orthogonality" to sets of functions, what information is the inner product revealing to us?

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# Orthogonality and physical applications

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