# Category:Weak-* Topologies

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This category contains results about **Weak-* Topologies**.

Definitions specific to this category can be found in Definitions/Weak-* Topologies.

Let $\GF \in \set {\R, \C}$.

Let $X$ be a topological vector space over $\GF$.

Let $X^\ast$ be the topological dual space of $X$.

For each $x \in X$, define $x^\wedge : X^\ast \to \GF$ by:

- $\map {x^\wedge} f = \map f x$

Let:

- $\sigma = \set {x^\wedge : x \in X}$

Let $w^\ast$ be the initial topology on $X^\ast$ with respect to $\sigma$.

We say that $w^\ast$ is the **weak-$\ast$ topology** of $X^\ast$.

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Weak-* Topologies"

The following 24 pages are in this category, out of 24 total.

### C

- Characterization of Continuity of Linear Functional in Weak-* Topology
- Characterization of Convergent Net in Weak-* Topology
- Characterization of Dual Operator
- Closed Unit Ball in Normed Dual Space is Weak-* Closed
- Closed Unit Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Metrizable
- Closed Unit Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Separable
- Closed Unit Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Sequentially Compact
- Closure in Weak-* Topology in terms of Annihilators

### E

### N

### V

### W

- Weak-* Closed Linear Subspace of Normed Dual Space is Isometrically Isomorphic to a Normed Dual Space
- Weak-* Dense Subset of Normed Dual Space Separates Points
- Weak-* Metrizability of Closed Unit Ball in Normed Dual of Normed Vector Space implies Original Space is Separable
- Weak-* Topology is Hausdorff