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Let $\{x_n\}$ and $\{y_n\}$ is the two biorthogonal sequences in the Hilbert space $H$. It means $<x_i,y_i>=\delta_{ij}$. Prove that the two sequences is linear independence.

My teacher said that is an easy exercise but i'm very thank you for answer me. Sorry for my bad english.

Hoàng
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1 Answers1

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Hint: Suppose $\alpha_1 x_1 + \ldots + \alpha_n x_n = 0$. What happens if you inner product both sides with $y_1$?

Theo Bendit
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  • i can see that the element of ${x_n}$ is linear independence, but it said that the two sequences is linear independence, so is it right if i take $\alpha_1{x_n}+\alpha_2{y_n}=0$ and then prove $\alpha_1=\alpha_2=0$? – Hoàng Nov 18 '17 at 12:12
  • No; you're trying to show that each sequence is linearly independent by themselves. So, whatever argument you did for the $\lbrace x_n \rbrace$ terms, do it for $\lbrace y_n \rbrace$. You don't have to show that they're independent of each other. Besides, I'm not even sure what $\alpha_1 \lbrace x_n \rbrace$ refers to! – Theo Bendit Nov 18 '17 at 12:16
  • okay thank you so much for answer ^^ – Hoàng Nov 18 '17 at 12:22