Proving the Bounded Linearity of A in l^{p} Space

[daishin]

Homework Statement

Let 1$$\leq$$p$$\leq$$$$\infty$$ and suppose ($$\alpha_{ij}$$ is a matrix such that (Af)(i)=$$\sum^{\infty}_{j=1}$$$$\alpha$$$$_{ij}$$f(j) defines an element Af of $$l^{p}$$ for every f in $$l^{p}$$. Show that A is a bounded linear functional on $$l^{p}$$

Homework Equations

Isn't this obvious if we apply theorem that says following are equivalent for A:X-->X a linear transformation on normed space?
(a)A is bounded linear functional
(b)A is continuous at some point
(c)There is a positive constant c such that ||Ax||$$\leq$$c||x|| for all x in X.

The Attempt at a Solution

Isn't the contiunuity of f obvious? So by the theorem, I think A is bounded linear functional on $$l^{p}$$. Could you guys correct me if I am wrong? Or if I am right could you just say it is right?

Thanks

 

[morphism]

Er, the continuity of f? f is an element of l^p, not a function. Also, how is A a functional on l^p? To me a functional on X is a mapping into the scalar field, and I believe this is standard terminology; A looks like a mapping from l^p to l^p (namely f $\mapsto$ Af).

Edit:
Looking at what you said again, I think you might've meant to say "continuity at f". If so, what f?

 

[brown042]

Maybe then I should say that A is bounded linear transformation?
But still isn't continuity of A obvious by construction?

 

[morphism]

Why is it obvious? Can you post your proof?

 

[daishin]

Sorry. It's not obvious. It seems continuous though.
I know for given epsilon > 0, I need to find delta>0 such that ||f||<delta implies
||Af||<epsilon. Hmm.. How can I find such delta? Or use cauchy sequence?

 

[morphism]

It's not that it's a difficult problem, but I just wouldn't say that the continuity of A is obvious. Anyway, I would use characterization (c) in the first post. Namely, try to compute ||Ax||_p given an arbitrary sequence x=(x_1, x_2, ...) in l^p. It will help to split this into three cases, depending on whether p=1, 1<p<$\infty$, or p=$\infty$. Holder's inequality (and Cauchy-Schwarz) will be helpful in the first two cases.

If you need any more hints, post back.

Also, a much more interesting (and more difficult!) problem is to prove that the same map, x $\mapsto$ Ax, is continuous if it maps sequence in l^p to sequences in l^p', where 1 < p, p' < $\infty$. You might want to try to tackle this one if you're looking for a challenge.