Suppose you have two UNIT vectors in some inner product space (V, <,>).

Then is the component of in the direction.

(Recall that , decomposes into parallel and perpendicular components w/resp to :

.)

Conceptually, should , some generic unit vector, have a greater component in the direction than itself does?

No.Conceptually, each unit vector "should have the greatest magnitude among all unit vector in its own direction." Right?

OK. So that means weexpect.... , with equality iff .

Thus weexpect.... , with equality iff .

Thus weexpect.... , with equality iff .

Thus weexpect.... , with equality iff .

Thus weexpectinner products of *unit* vectors to take values in [-1,1], with it being only when one vector is the other.

Now, suppose and are any two non-zero vectors (CS obviously holds if one of the vectors is ). What do we expect?

Consider the proceeding in light of their unit vectors and :

Weexpect.... , with equality iff , i.e. iff .

So weexpect.... , with equality iff = some multiple of .

Thus weexpectthe Cauchy-Schwarz inequality to hold - because we expect inner products of unit vectors to take values in [-1,1].

That's its conceptual meaning to me.