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For any complex inner product space $V$ and for any $u,v \in V$, we have:

$$4 \langle u,v \rangle = \| u+v\| ^2 - \| u-v\|^2 +i\| u+iv\|^2 -i\| u-iv\|^2$$

to which my lecture notes conclude: "any inner product is determined by the lengths of vectors."

I do not understand what the statement means. In particular, I have tried to use the following example numerically and graphically, but I still do not quite get it.

Is $u$ or $v$ a vector with complex numbers as co-ordinates? If so, I still don't quite get what is meant by "determined by the length of vectors". Does it mean adding up the length of each real and imaginary part and then adding together?

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  • $\begingroup$ Ask the lecturer/professor who provided the notes! I'm sure they would be happy to clarify. $\endgroup$ Sep 12, 2023 at 8:51
  • $\begingroup$ It means you cannot have two different inner produtcs $\langle x, y \rangle$, $\langle x, y \rangle'$ such that $\langle x, x \rangle=\langle x, x \rangle'$ for all $x$. $\endgroup$ Sep 12, 2023 at 8:51
  • $\begingroup$ You have written down the "Polarisation identity", cf Polarization identity in Hilbert space or Pre-Hilbert space and polarization identities . $\endgroup$
    – Hanno
    Sep 12, 2023 at 9:10
  • $\begingroup$ Would you consider to accept an answer if it fits your bill? $\endgroup$
    – Hanno
    Sep 19, 2023 at 9:53

2 Answers 2

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It's not 100% to say that in a complex inner product space, a vector has complex numbers as coordinates. But I suppose that at the current level of your understanding of this material, you may think so and it's not too bad. So, yeah, let's say that a vector has complex number as coordinates.

A vector $v$'s norm, denoted as $\|v\|$, can also be called its length (the distance from zero to this vector). For two vectors $u$ and $v$, the norm of their difference, $\|u-v\|$, is the distance between them. From the equation you wrote, we can see that the value of inner product between two vectors $u$ and $v$ is determined by lengths of 4 vectors: $u+v, u-v, u+iv, u-iv$. So, if you know these 4 lengths, then you know what the inner product equals. And also, it's impossible to define two different inner products on the same complex vector space, so that they produce different values of the inner product but the same lengths for all vectors.

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    $\begingroup$ I like your answer, and I'd propose to make the last sentence more succinct: ... different values of the inner product ... the same lengths for all vectors. $\endgroup$
    – Hanno
    Sep 12, 2023 at 12:33
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The statement "any inner product is determined by the lengths of vectors." means, that it is possible to define the inner product solely by the norm.

Instead of calculating the inner product $\langle u,v\rangle$ it is possible to calculate the right-hand side.

It is not of importance if the values are real or imaginary of $ u $ and $ v $.

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