Weak Convergence in Normed Dual Space of Reflexive Normed Vector Space is Equivalent to Weak-* Convergence

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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

$\blacksquare$


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