Hi, I would really appreciate any hints towards a solution to the following prob

Question

Hi,

I would really appreciate any hints towards a solution to the following problem. It goes like this:

Let x be a vector in R^k

Show that ..:

if a constant c > 0 exists, such that x*y <= c ||y|| for all y in R^k

then ||x|| <= c

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Im pretty sure the problem should be solvable, using no more than the following vector properties:

Cauchy-Schwarz:

|x*y| <= ||x|| ||y|| for all x,y in R^k

the "triangle inequality" for vectors:

||x+y|| <= ||x|| + ||y||

and the following upper bound on the length of a vector x in R^k:

||x|| <= sqrt(k)*max(|x_i|) , for i in 1..k

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I´ve tried:

- the straightforward method of manipulating cauchy-schwarz and the given inequality x*y <= c ||y|| , but dont know how to establish any upper bound on ||x|| from that.

- using the mentioned upperbound on ||x|| ( ||x|| <= sqrt(k)*max(|x_i|) , for i in 1..k ) , and doing the manipulations by writing out the vectors as sums of their coordinates.

- showing the contrapositive, that c > ||x|| implies x*y > c ||y||

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I would really appreciate help with this!

Explanation / Answer

Let y = x.

Then, x*y = ||x||^2

Yet, x*y <= c ||y|| = c ||x||

Then, ||x||^2 <= c||x||.

Then, as ||x|| >= 0, if ||x|| = 0, ||x|| < c, as c > 0

If ||x|| > 0, then, dividing by ||x||, as ||x||^2 <= c||x||,

||x|| <= c

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