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The Contragradient or Dual Representation is defined as:

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My trail: $((R+S)'(g)f)(x) = f((R +S)(g)^{-1} x)$, $(g \in G, f \in V', x \in V)$

But then I am stucked, how to distribute the inverse, Is it allowed to write it as $f ((R^{-1} + S^{-1})(g)x)$ ...... if I can do so then the solution will be very easy ..... so if anyone give me a hint about the solution I would appreciate this very much?

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    $\begingroup$ Transpose is linear. $\endgroup$ – David Hill Feb 13 at 5:16
  • $\begingroup$ could you please say a little more details?@DavidHill $\endgroup$ – hopefully Feb 15 at 0:05
  • $\begingroup$ @DavidHill but my definition for duality does not contain transpose ...... from where the transpose will come? $\endgroup$ – hopefully Feb 16 at 2:14

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