He writes, talking about the ket space X of a physical system,(adsbygoogle = window.adsbygoogle || []).push({});

In general, the dual space [itex]X^*[/itex] and the state space [itex]X[/itex] are not isomorphic, except of course, if [itex]X[/itex] is finite dimensional.

Can someone explain why they are isom. in the event of finite dimensionality?

For instance, say X is finite. To every |x> in X is associated the linear functional [itex]<x|[/itex] (the "inner product functional"). But also, given any (non-identity) linear operator A on X, <x|A is a new linear functionnal on X, is it not*? So that makes the cardinality of [itex]X^*[/itex] greater than the cardinality of X, so there can be no bijection between them.

What's wrong?

*I should saypotentiallynew, because if A maps every |x1> towards |x2>'s such that, by miracle, <x|x1>=<x|x2>, then <x| and <x|A really are the same functional...

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# Statement in Cohen-Tanoudji I don't understand

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