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I know that given a bilinear form $B$ and it's corresponding quadratic form $Q$, assuming that $B$ is complex and bounded, that $\|B\| \le 2\|Q \|$

However, I've failed to manufacture an example of a bilinear form $B$ where this inequality is strict.

Help?

Shai Deshe
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    What is the definition of $||Q||$? – Davide Giraudo Feb 04 '12 at 11:23
  • Take $B$ to be ordinary multiplication on $\mathbb C^1$. Then $\lVert B\rVert=\lVert Q\rVert=1\lneq2=2\lVert Q\rVert$. – mr_e_man Jun 01 '23 at 20:46
  • Or did you want the opposite of a strict inequality? For an example where it's an equality $\lVert B\rVert=2\lVert Q\rVert$, see a comment here https://math.stackexchange.com/questions/4710809/sesquilinearform-only-small-on-diagonal – mr_e_man Jun 01 '23 at 20:47

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