- #1
Chain
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If a sequence of operators [itex]\{T_n\}[/itex] converges in the norm operator topology then:
$$\forall \epsilon>0$$ $$\exists N_1 : \forall n>N_1$$ $$\implies \parallel T - T_n \parallel \le \epsilon$$
If the sequence converges in the strong operator topology then:
$$\forall \psi \in H$$ $$\forall \epsilon>0$$ $$\exists N_2 : \forall n>N_2$$ $$\implies \parallel T\psi - T_n\psi \parallel \le \epsilon$$
Where H is the Hilbert space that the operators act on. I believe that norm convergence implies strong convergence since for [itex]n>N_1[/itex]:
$$\parallel T\psi - T_n\psi \parallel = \parallel(T-T_n)\psi \parallel \le \parallel T - T_n \parallel \parallel \psi \parallel \le \epsilon \parallel \psi \parallel $$
Since the magnitude of any vector in the Hilbert space must be finite we can scale [itex]\epsilon[/itex] so that the RHS of the inequality is arbitrarily close to zero hence we have convergence in the strong operator topology.
However in my functional analysis book (methods of mathematical physics by Simon and Reed) it says that the map taking an operator to its adjoint is continuous in the norm topology but not the strong topology. This means if a sequence converges to an operator T then the sequence obtained by taking the adjoint of every operator in the original sequence also converges to some operator. Hence we have a sequence converging in the norm topology but not in the strong topology.
I would be very grateful if someone could point out my mistake (or perhaps a mistake in Simon and Reed however this is not the only example of norm convergence does not imply strong convergence that I have seen).
$$\forall \epsilon>0$$ $$\exists N_1 : \forall n>N_1$$ $$\implies \parallel T - T_n \parallel \le \epsilon$$
If the sequence converges in the strong operator topology then:
$$\forall \psi \in H$$ $$\forall \epsilon>0$$ $$\exists N_2 : \forall n>N_2$$ $$\implies \parallel T\psi - T_n\psi \parallel \le \epsilon$$
Where H is the Hilbert space that the operators act on. I believe that norm convergence implies strong convergence since for [itex]n>N_1[/itex]:
$$\parallel T\psi - T_n\psi \parallel = \parallel(T-T_n)\psi \parallel \le \parallel T - T_n \parallel \parallel \psi \parallel \le \epsilon \parallel \psi \parallel $$
Since the magnitude of any vector in the Hilbert space must be finite we can scale [itex]\epsilon[/itex] so that the RHS of the inequality is arbitrarily close to zero hence we have convergence in the strong operator topology.
However in my functional analysis book (methods of mathematical physics by Simon and Reed) it says that the map taking an operator to its adjoint is continuous in the norm topology but not the strong topology. This means if a sequence converges to an operator T then the sequence obtained by taking the adjoint of every operator in the original sequence also converges to some operator. Hence we have a sequence converging in the norm topology but not in the strong topology.
I would be very grateful if someone could point out my mistake (or perhaps a mistake in Simon and Reed however this is not the only example of norm convergence does not imply strong convergence that I have seen).