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 1
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-$\ast$.
$\blacksquare$
Proof 2
Let $f \in X^\ast$ be the limit of $\sequence {f_n}_{n \mathop \in \N}$, i.e.:
- $\norm {f_n - f}_{X^\ast} \stackrel{n \to \infty}{\longrightarrow} 0$
Thus, for each $x \in X$:
\(\ds \size {\map {f_n} x - \map f x}\) | \(=\) | \(\ds \size {\map {\paren {f_n - f} } x}\) | Definition of Vector Space of Bounded Linear Functionals | |||||||||||
\(\ds \) | \(\le\) | \(\ds \norm {f_n - f}_{X^\ast} \norm x_X\) | Fundamental Property of Norm on Bounded Linear Functional | |||||||||||
\(\ds \) | \(\) | \(\ds \stackrel{n \to \infty}{\longrightarrow} 0\) |
$\blacksquare$
Sources
- 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $27.4$: Weak-$\ast$ Convergence