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For complex numbers $a_{n}, b_{n}$, what would be the next step to simplify this expression:

$$\left \|\sum_{n=1}^\infty a_{n} b_{n}\right \|^{2}=\left \langle \sum_{n=1}^\infty a_{n}b_{n},\sum_{m=1}^\infty a_{m}b_{m}\right \rangle$$

Is it $=\sum_{n=1}^\infty \sum_{m=1}^\infty |a_{n}|^{2}|b_{m}|^{2}$ !?

where $|a_{n}|^{2}=a_{n}\overline{a_{n}}$

  • 1
    When you say "Is it = $\sum_{n=1}^\infty \sum_{n=1}^\infty |a_{n}|^{2}|b_{n}|^{2}$", the two sums should use different indices.2011-03-16
  • 0
    Right, I fixed it.2011-03-16

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