## Dual of a well known space

Prove that , where .

Prove that, in general, .

**Solution:** If is a Banach space where is a vector space over the field and if , then we denote .

If then define . This functional is obviously linear, and from Holder’s inequality, we get that . This implies that the series from the definition of is absolutely convergent, and since is a Banach space, the series is convergent, therefore is well defined for any , and , which implies that .

Therefore, we can define by .

It is obvious that is linear. If we assume that , then taking where the ‘th position is and the rest are ‘s it follows that and , therefore , and . This implies that is injective.

Now, we pick and we see that . We should prove that .

For this, we consider which is in since all the terms of the sequence are except a finite number of them. .

We have , therefore . Taking , we get that and . Therefore , and thus, is surjective. This means that is indeed an isomorphism, and . Conversely, , by Holder’s inequality, therefore . Therefore, is an isometry also.