A Comparison of the Weak and Weak* Topologies

# A Comparison of the Weak and Weak* Topologies

We will now compare the various topologies on $X$ and $X^*$.

## Topologies on X

Topology Alternate Names Description
The Norm Topology on $X$ The Strong Topology on $X$ The topology obtained from the norm $\| \cdot \|_X$ on $X$. It is generated by the collection of open balls $B_X(x, r) = \{ y : \| x - y \|_X < r \}$.
The Weak Topology on $X$ The Weak Topology on $X$ induced by $X^*$ The weakest topology which makes every $f \in X^*$ weakly continuous. A sequence of points $(x_n) \subset X$ weakly converges to $x \in X$ if and only if $\displaystyle{\lim_{n \to \infty} f(x_n) = f(x)}$ for every $f \in X^*$.
(1)
\begin{align} \quad \mathrm{weak \: topology \: on \:} X \subseteq \mathrm{norm \: topology \: on \:} X \end{align}
 Proposition 1: Let $X$ be a normed linear space. Then: a) If $O$ is weakly open in $X$ then $O$ is norm open in $X$. b) If $C$ is weakly closed in $X$ then $C$ is norm closed in $X$. c) If $K$ is norm compact in $X$ then $K$ is weakly compact in $X$. d) If $f : X \to \mathbb{R}$ is weakly continuous on $X$ then $f$ is norm continuous on $X$. e) If $(x_n)$ norm converges to $x$ then $(x_n)$ weakly converges to $x$.

## Topologies on X^*

Topology Alternate Names Description
The (Operator) Norm Topology on $X^*$ The Strong Topology on $X^*$ The topology obtained from the norm $\| \cdot \|_{\mathrm{operator}}$ on $X^*$. It is generated by the collection of open balls $B_{X^*} (f, r) = \{ g : \| f - g \|_{\mathrm{operator}} < r \}$.
The Weak Topology on $X^*$ The Weak Topology on $X^*$ induced by $X^{**}$ The weakest topology which makes every $\varphi \in X^{**}$ weakly continuous. A sequence of functionals $(f_n) \subset X^*$ weakly converges to $f \in X^*$ if and only if $\displaystyle{\lim_{n \to \infty} \varphi(f_n) = \varphi(f)}$ for every $\varphi \in X^{**}$.
The Weak-* Topology on $X^*$ The Weak Topology on $X$ induced by $\hat{X}$ or $J(X)$ The weakest topology which makes every $\hat{x} \in \hat{X}$ weak-* continuous. A sequence of functionals $(f_n) \subset X^*$ weak-* converges to $f \in X^*$ if and only if $\displaystyle{\lim_{n \to \infty} f_n(x) = f(x)}$ for every $x \in X$.
(2)
\begin{align} \quad \mathrm{weak-* \: topology \: on \:} X^* \subseteq \mathrm{weak \: topology \: on \:} X^* \subseteq \mathrm{operator \: norm \: topology \: on \:} X^* \end{align}
 Proposition 2: Let $X$ be a normed linear space. Then: a) If $O$ is weak-* open in $X^*$ then $O$ is weakly open in $X^*$. If $O$ is weakly open in $X^*$ then $O$ is operator norm (strongly) open in $X^*$. b) If $C$ is weak-* closed in $X^*$ then $C$ is weakly closed in $X^*$. If $C$ is weakly closed in $X^*$ then $C$ is operator norm (strongly) closed in $X^*$. c) If $K$ is norm (strongly) compact in $X^*$ then $K$ is weakly compact in $X^*$. If $K$ is weakly compact in $X^*$ then $K$ weak-* compact in $X^*$. d) If $f : X^* \to \mathbb{R}$ is weak-* continuous on $X$ then $f : X^* \to \mathbb{R}$ is weakly continuous on $X$. If $f : X^* \to \mathbb{R}$ is weakly continuous on $X$ then $f : X^* \to \mathbb{R}$ is operator norm (strongly) continuous on $X$. e) If $(f_n)$ norm converges to $f$ then $(f_n)$ weakly converges to $f$. If $(f_n)$ weakly converges to $f$ then $(f_n)$ weak-* converges to $f$.

Propositions 1 and 2 above follow immediately from the proposition proved on the Weaker and Stronger Topologies page.