diff options
Diffstat (limited to 'pages/more-functional-analysis')
| -rw-r--r-- | pages/more-functional-analysis/topological-vector-spaces/index.md | 2 | ||||
| -rw-r--r-- | pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md | 9 |
2 files changed, 5 insertions, 6 deletions
diff --git a/pages/more-functional-analysis/topological-vector-spaces/index.md b/pages/more-functional-analysis/topological-vector-spaces/index.md index 745d53b..a3e6220 100644 --- a/pages/more-functional-analysis/topological-vector-spaces/index.md +++ b/pages/more-functional-analysis/topological-vector-spaces/index.md @@ -49,7 +49,7 @@ A subset $A \subset X$ is said to be {% theorem %} These properties of subsets of $X$ are stable under arbitrary intersections. -Thus we obtain the notions of +Thus, we obtain the notions of *convex hull* $\co A$, *balanced hull* $\bal A$, and *absolutely convex hull* $\aco A$. diff --git a/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md b/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md index 277ecd3..ea0b22b 100644 --- a/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md +++ b/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md @@ -29,7 +29,7 @@ We say that the bilinear form $b : V \times W \to \KK$ is *nondegenerate*, if it $$ \begin{gather*} \forall v \in V : \qquad ( \forall w \in W : \angles{v,w} = 0 ) \implies v = 0 \\ -\forall w \in W : \qquad ( \forall v \in V : \angles{v,w} = 0 ) \implies w = 0 +\forall w \in W : \qquad ( \forall v \in V : \angles{v,w} = 0 ) \implies w = 0 \end{gather*} $$ @@ -45,7 +45,6 @@ $$ Then $b$ is nondegenerate if and only if both $c$ and $\tilde{c}$ are injective. - {% definition Dual Pair %} A *dual pair* (or *dual system* or *duality*) $\angles{V,W}$ over a field $\KK$ is constituted by two vector spaces $V$ and $W$ over $\KK$ @@ -56,9 +55,9 @@ and a nondegenerate bilinear form $\angles{\cdot,\cdot} : V \times W \to \KK$. {% definition Weak Topology %} Suppose $\angles{X,Y}$ is a dual pair of vector spaces over a field $\KK$. -We define the *weak topology on $X$*, denoted by $\sigma(X,Y)$, as -the [initial topology](/pages/general-topology/universal-constructions.html#initial-topology) induced by the maps -$\angles{\cdot,y} : X \to \KK$, where $y \in Y$. +We define the *weak topology on $X$*, denoted by $\sigma(X,Y)$, as the +[initial topology](/pages/general-topology/universal-constructions.html#initial-topology) +induced by the maps $\angles{\cdot,y} : X \to \KK$, where $y \in Y$. Similarly, the *weak topology on $Y$*, denoted by $\sigma(Y,X)$, is the initial topology induced by the maps $\angles{x,\cdot} : Y \to \KK$, where $x \in X$. |