Convergence in Normed Dual Space implies Weak-* Convergence

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Theorem

Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space.

Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.

Let $\sequence {f_n}_{n \mathop \in \N}$ be a convergent sequence in $X^\ast$.


Then $\sequence {f_n}_{n \mathop \in \N}$ converges weakly-$\ast$.


Proof

From Convergent Sequence in Normed Vector Space is Weakly Convergent, $\sequence {f_n}_{n \mathop \in \N}$ converges weakly.

From Weakly Convergent Sequence in Normed Dual Space is Weakly-* Convergent, $\sequence {f_n}_{n \mathop \in \N}$ converges weakly-$\ast$.

$\blacksquare$


Sources