Let $$X=l_p^{(3)}$$, where $$1\lt p \lt \infty$$, and $$\phi(x) = x_1-2x_2+3x_3$$.
Decide whether $$\phi$$ is bounded, and if so, find $$||\phi||$$.
So by marking $$y=(1,-2,3)$$, we can see that $$\phi(x)=\sum_{j=1}^3x_iy_i$$.
Therefore, by Riesz representation theorem, we get that $$||\phi||=||y||=(1+2^p+3^p)^{\frac{1}{p}}$$.
However, in the book that I study, the final answer is $$(1+2^q+3^q)^{\frac{1}{q}}$$, where $$1\lt q\lt \infty$$ and $$\frac{1}{p}+\frac{1}{q}=1$$.
I know that $$l_p^*=l_q$$... But why $$y\in l_q$$ and not $$y\in l_p$$?