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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 $y\in l_p$?

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