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Solution from the answer book

So I got stuck trying to prove the Schwarz inequality by expressing the product of magnitudes as a sum of two squares, and was reading the solution from the answer book; however I could not understand the last part on "the difference", i.e. $ \sum_{i≠j}({x_i}^2{y_j}^2 - x_iy_ix_jy_j) = 2 \sum_{i<j} ({x_i}^2{y_j}^2 + {x_j}^2{y_i}^2 - x_iy_ix_jy_j) $. Please help me understand how the terms doubled and why the next line suddenly had a square term.

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There are two typos in this page: On the first line after "The difference...", the "2" should be in front of $x_iy_ix_jy_j$, the reason being that when you change the index from $i\neq j$ to $i<j$ you have to double all the terms: say on the left hand side you have both $x_1^2y_2^2$ and $x_2^2y_1^2$, and since on the right hand side $I<j$ you have to include the second one. The same applies to the products $x_iy_ix_jy_j$. Each of the terms is obviously a square. There is a typo on the following line: there is no "2" in front of the sum, as in the original (and correct) statement of the inequality.

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  • $\begingroup$ I see, so it was two typos at once! That makes much more sense. $\endgroup$
    – MountainC
    Feb 7 '20 at 7:41
  • $\begingroup$ Yes. Two typos at once is very unlikely assuming they follow a Poisson distribution... but in this case the typos are linked to each other so probably it is not safe to assume that for a Math book. Spivak's books are great anyways. $\endgroup$
    – GReyes
    Feb 7 '20 at 15:29

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