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Let $\{X_i:i\in I\}$ be a collection of normed spaces. If $1\leq p < \infty$, show that the dual space of $\bigoplus_p X_i$ is isometrically isomorphic to $\bigoplus_q {X_i}^\ast$, where $1/p+1/q=1$.

I know that if X is normed space, then $X^\ast$ is Banach space.

Also, the product of finite number of Banach spaces is Banach space.

Can you give me some hints?

asfajaf
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  • Is your $\oplus_p$ just the direct sum and $p$-norm? Ie are the elements of $\oplus_p X_i$ finite linear combinations of the elements of $X_i$ and the norm just $|\sum_i x_i|:=\left(\sum_i|x_i|_i^p \right)^{1/p}$ or is there something more general happening? – s.harp May 07 '16 at 12:35
  • I think that this problem is general case of l_p – asfajaf May 07 '16 at 21:01
  • Can you expand on what your sum is? Because, note that if it is just the direct sum with the norm described as above, then a sum of such spaces need not be complete, even if every space in the sum is complete. This would make it impossible for $\oplus X_i^*$ to be the dual of a normed space if $I$ is infinite. – s.harp May 08 '16 at 12:37

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