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Does the function below define an inner product? $$\langle (x, y), (z, t)\rangle = xz − yt$$

I know how to prove it given two vectors (e.g. $\langle(x,y),(z,t)\rangle$) demonstrating symmetry, linearity and positive-definiteness. But it's not clear to me how to deal with functions.

Should I find $u,v$ (e.g. $u=(x,y), z=(z,-t)$) and demonstrate the properties above?

Another example, a more complex one that would be hard to find $u,v$: $$\langle (x,y),(z,t)\rangle = 2xz -xt -yz +2yt$$

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You need to demonstrate or refute the properties of an inner product for $$ \langle (x,y) , (z,t)\rangle = xz -yt$$

If you suspect that it is not an inner product then yes, you need to find suitable vectors that violate one of the properties of an inner product.

If $(x,y), (z,t)$ are vectors in $\mathbb R^2$ then pick for example $(x,y) = (z,t) = (1,2)$ and note that $\langle (1,2) , (1,2) \rangle = 1-4 < 0$ so that positive-definiteness is violated and it cannot be an inner product.

For the other example you give you need to proceed in the same manner: first establish whether you want to prove or refute it and then either try to find counterexamples or try to find a proof.

For a proof you need to verify the following four properties of inner products:

(i) Symmetry: $\langle x,y\rangle = \langle y,x\rangle$

(ii) Linearity: $\langle \alpha x + \beta x',y\rangle = |\alpha|\langle x ,y\rangle + |\beta| \langle x',y\rangle$

(iii) $\langle x,x\rangle \ge 0$

(iv) $\langle x,x \rangle = 0$ iff $x=0$

Example (symmetry):

$\langle (x,y),(z,t)\rangle = 2xz -xt -yz +2yt$

$\langle (z,t), (x,y) \rangle = 2xz - yz -xt + 2yt$

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  • $\begingroup$ Hi! Thanks for the answer. Now it's clear to me how to refute it, but still not clear how to proof. It seems that the second function is an inner product space. If so, could you help me how to proof it? If I get one example I think I can solve the others on my own. $\endgroup$ – Ricardo Mayerhofer Jan 26 '14 at 14:12
  • $\begingroup$ @RicardoMayerhofer I edited my answer. I hope this helps. $\endgroup$ – user89987 Jan 27 '14 at 13:20

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