Determine the norm $~~\|\phi\|,~~$ where $~~\phi \in l_p^*$.

Question

Let us consider the normed space $~~X=l_p.~~$ Now consider an element $~~\phi \in X^*~~$ given by $$\phi(x)=\sum_{k=1}^N\frac{x_k}{N^{\frac{1}q}},~~~~~\text{ where } x=(x_1,x_2,\cdots,x_n,\cdots) \in X.$$ Also note that $~~N \in \mathbb N,~~~1<p,~q<\infty~~$ and $~~\dfrac{1}p+\dfrac{1}q=1.$ Now determine the norm $~~\|\phi\|$.
Now by definition of $~~l_p~~$ space we have $$\|x\|_{l_p} = \left(\sum_{k=1}^{\infty} |x_k|^p\right)^{\frac{1}p},~~~~\text{ for }~~x=\left(x_k\right)_{k \ge 1} \in l_p.$$ Now notice that, by definition $$\|\phi\|=\sup \left\{|\phi(x)|~:~\|x\|_{l_p}=1\right\},$$ and I am totally stuck here. Can you please help me to solve this, or how to find the norm.

Thanks for your time.

Answer 1

We have $$|\phi (x) | \leq \sum_{k=1}^N\frac{|x_k|}{N^{\frac{1}{q}}}=\frac{1}{N^{\frac{1}{q}}}\sum_{k=1}^N |x_k|\leq \frac{1}{N^{\frac{1}{q}}}\left(\sum_{k=1}^N |x_k|^p \right)^{\frac{1}{p}}\cdot \left(\sum_{k=1}^N 1^q \right)^{\frac{1}{q}}\leq||x||_{\ell^p}$$

Hence $$||\phi ||\leq 1.$$ But if you take $x_0= (N^{\frac{1}{q} -1},N^{\frac{1}{q} -1},....,N^{\frac{1}{q} -1}, 0,0,...)$ then $||x_0||_{\ell^p} =1$ and $$\phi (x_0 )=1 $$ therefore $$||\phi ||=1.$$