Weak Convergence in Normed Dual Space of Reflexive Normed Vector Space is Equivalent to Weak-* Convergence
Theorem
Let $\mathbb F$ be a subfield of $\C$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a reflexive normed vector space over $\mathbb F$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $f \in X^\ast$.
Then:
- $\sequence {f_n}_{n \mathop \in \N}$ converges weakly to $f$ if and only if $\sequence {f_n}_{n \mathop \in \N}$ converges weakly-$\ast$ to $f$.
Proof 1
Sufficient Condition
Suppose that $\sequence {f_n}_{n \mathop \in \N}$ converges weakly to $f$.
From Weakly Convergent Sequence in Normed Dual Space is Weakly-* Convergent, we have:
- $\sequence {f_n}_{n \mathop \in \N}$ converges weakly-$\ast$ to $f$.
$\Box$
Necessary Condition
Suppose that $\sequence {f_n}_{n \mathop \in \N}$ converges weakly-$\ast$ to $f$.
Then:
- $\map {f_n} x \to \map f x$
for each $x \in X$.
We aim to show that:
- $\map F {f_n} \to \map F f$ for all $F \in X^{\ast \ast}$.
Let $F \in X^{\ast \ast}$.
Since $X$ is reflexive, we have:
- for each $F \in X^{\ast \ast}$ there exists $x_F \in X$ such that $F = x_F^\wedge$
where $x_F^\wedge : X^\ast \to \mathbb F$ is defined by:
- $\map {x_F^\wedge} f = \map f x$
for each $f \in X^\ast$.
Then, we have:
- $\map {f_n} {x_F} \to \map f {x_F}$
That is:
- $\map {x_F^\wedge} {f_n} \to \map {x_F^\wedge} f$
Since $F = x_F^\wedge$, we therefore have:
- $\map F {f_n} \to \map F f$
Since $F \in X^{\ast \ast}$ was arbitrary, we have:
- $\sequence {f_n}_{n \mathop \in \N}$ converges weakly to $f$ in $X^{\ast \ast}$.
$\blacksquare$
Proof 2
Let $J : X \to X^{**}$ be the evaluation linear transformation.
In view of Definition of Reflexive Space, $J$ is especially a surjection.
Therefore:
\(\ds f_n\) | \(\weakconv\) | \(\ds f\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \forall x'' \in X^{**}: \, \) | \(\ds \map {x''} {f_n}\) | \(\to\) | \(\ds \map {x''} f\) | Definition of Weak Convergence | |||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \forall x \in X: \, \) | \(\ds \map {\map J x} {f_n}\) | \(\to\) | \(\ds \map {\map J x} f\) | as $J : X \to X^{**}$ is a surjection | |||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \forall x \in X: \, \) | \(\ds \map {f_n} x\) | \(\to\) | \(\ds \map f x\) | Definition of $J$ | |||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds f_n\) | \(\weakstarconv\) | \(\ds f\) | Definition of Weak-* Convergence |
$\blacksquare$
Sources
- 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $27.4$: Weak-$\ast$ Convergence