In the book "Linear Functional Analysis" from Epstein there is a proof (page 116) of the following theorem:
The dual space of Lp is equal to Lq.
Later, it is stated as a theorem without proof that
for 1 <= p < +infinite, The dual space of lp is equal to lq.
I would like to prove this later theorem. I am able to find a member of the dual space of lp for each member of lq, namely
Let b = (b1, b2, b3, ...) be any member of lq; and let a = (a1, a2, a3, ...) be a member of lp.
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 lp (that is, belongs to the dual space of lp). Moreover, this correspondence between (lp)* and lq is norm-preserving.
Now, given a member l of the dual space of lp, I need to find a member b of lq 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,Hi Hoda,
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.Doug