Content deleted Content added
m Duplicate word removed |
m Disambiguating links to Separability (link changed to Separable space) using DisamAssist. |
||
Line 242:
The weak-topology on {{math|L(''X''; ''Y'')}} has the following properties:
<ul>
<li>If {{mvar|X}} is [[Separable space|separable]] (i.e. has a countable dense subset) and if {{mvar|Y}} is a metrizable topological vector space then every equicontinuous subset {{mvar|H}} of {{math|L<sub>𝜎</sub>(''X''; ''Y'')}} is metrizable; if in addition {{mvar|Y}} is separable then so is {{mvar|H}}.{{sfn | Schaefer | 1999 | p=87}}
* So in particular, on every equicontinuous subset of {{math|L(''X''; ''Y'')}}, the topology of pointwise convergence is metrizable.</li>
<li>Let {{math|''Y''<sup>''X''</sup>}} denote the space of all functions from {{mvar|X}} into {{mvar|Y}}. If {{math|L(''X''; ''Y'')}} is given the topology of pointwise convergence then space of all linear maps (continuous or not) {{mvar|X}} into {{mvar|Y}} is closed in {{math|''Y''<sup>''X''</sup>}}.
| |||