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if we supposexandyare two elements of some vector spaceV(say ℝ^{n}), and if we consider a linear functionf:V→V', we know that the inner product of the transformed vectors is given by: [tex]\left\langle f\mathbf{x} , f\mathbf{y} \right\rangle = \left\langle \mathbf{x} , \overline{f}f\mathbf{y} \right\rangle = \left\langle \overline{f}f\mathbf{x} , \mathbf{y} \right\rangle[/tex] where [itex]\overline{f}[/itex] is the adjoint operator of [itex]f[/itex].

What can we say about [itex]\left\langle f\mathbf{x} , f\mathbf{y} \right\rangle[/itex] whenfis non-linear, for example adiffeomorphism?

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# How the inner product changes under non-linear transformation

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