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There is an example in the lecture notes I'm currently reading, in a chapter on the dual pairing of a topology, that in $E:=\ell^2$ the convergence of a sequence $(x_n)_n$ to zero in the $2$-norm-topology implies the convergence to zero in the $\sigma(E,E')$ topology. ($E'$ denotes the topological dual of $E$).

(To see how this topology is defined, please consult this wikipedia entry.)

Can someone give me an idea for a counterexample of a componentwise sequence, such that it does not converge in the $\sigma(E,E')$ topology ?

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    If $f$ is (norm) continuous it is (norm) sequentially continuous. The converse does not hold. The sequence of standard unit vectors converges weakly to the zero vector but does not converge in norm to the zero vector. – David Mitra May 13 '14 at 11:35
  • @DavidMitra Ok, that gave me almost all I needed. Would it be true that there are sequences $(x_n)$ in $\ell^2$ that converge componentwise but not in $\sigma(E,E')$ ? –  May 13 '14 at 12:56
  • Yes. Take $x_n=\sum_{i=1}^n e_i$. If the sequence is bounded in norm, this isn't possible. See this. – David Mitra May 13 '14 at 13:05

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