need to know insight concept or underlaying concept of Cauchy–Schwarz inequality

CAN ANYONE PLEASE TELL ME ABOUT

Cauchy–Schwarz inequality

1. WHAT IT MEANS ACTUALLY I KNOW FORMAL DEFINITION BUT I WANA INSIGHT CONCEPT WHY PRODUCT OF INDIVIDUAL VECTORS IS ALWAYS> AND EQUAL THAN INNER PRODUCT

2. SPECIFY THE CASE WHEN PRODUCT OF MAGNITUDES OF IND.VECTORS IS = MAG OF INNER PRODUCT

MAY THIS HELPS ME TO GET INSIGHT CONCEPT

I AM NEW TO THESES FORUMS IF ANYONE CAN EMAIL ME IT WILL BE GREAT FOR ME...

THANKX.

E-MAIL ADRESS:

junaidanwar194@gmail.com

Re: need to know insight concept or underlaying concept of Cauchy–Schwarz inequality

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 we __expect__.... , with equality iff .

Thus we __expect__.... , with equality iff .

Thus we __expect__.... , with equality iff .

Thus we __expect__.... , with equality iff .

Thus we __expect__ inner 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 :

We __expect__.... , with equality iff , i.e. iff .

So we __expect__.... , with equality iff = some multiple of .

Thus we __expect__ the 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.