Name: Hoda

Question:
Hi there!

In the book "Linear Functional Analysis" from Epstein there is a proof (page 116) of the following theorem:

The dual space of L^p is equal to L^q.

Later, it is stated as a theorem without proof that

for 1 <= p < +infinite, The dual space of l^p is equal to l^q.

I would like to prove this later theorem. I am able to find a member of the dual space of l^p for each member of l^q, namely

Let b = (b1, b2, b3, ...) be any member of l^q; and let a = (a1, a2, a3, ...) be a member of l^p.

then the linear functional defined as

my_l(a) = Sum of the infinite series

a1+b1, a2+b2, a3+b3, ...

is a bounded linear functional on l^p (that is, belongs to the dual space of l^p). Moreover, this correspondence between (l^p)* and l^q is norm-preserving.

Now, given a member l of the dual space of l^p, I need to find a member b of l^q such that l = my_l.

Could you please help me to finish this proof or point me to a book where I could find the solution? Thanks a lot in advance,

Several books have this result. For example, Page 143 of A.E. Taylor and D.C. Lay, Introduction to Functional Analysis, 2nd Edition, John Wiley & Sons, 1980.