Topologies on spaces of linear maps: Difference between revisions

Revision as of 20:05, 30 June 2022

Revision as of 20:11, 30 June 2022

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If <math>N</math> is [[Balanced set|balanced]]{{sfn|Narici|Beckenstein|2011|pp=371-423}} (respectively, [[Convex set|convex]]) then so is <math>\mathcal{U}(G, N).</math>

The equality

<ul>

<limath>\mathcal{U}(\varnothing, N) = F</math>
always holds.
If <math>s</math> is a scalar then <math>s \mathcal{U}(G, N) = \mathcal{U}(G, s N),</math> so that in particular, <math>- \mathcal{U}(G, N) = \mathcal{U}(G, - N).</math>{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>

Moreover,{{sfn|Narici|Beckenstein|2011|pp=19-45}}

<li><math>\mathcal{U}(G \cup H, M \cap N) \subseteq \mathcal{U}(G, M) \cap \mathcal{U}(H, N)</math> for any subsets <math>G, H \subseteq X</math> and any non-empty subsets <math>M, N \subseteq Y.</math>{{sfn|Jarchow|1981|pp=43-55}} For such subsets, it follows that,

<li><math display=block>\mathcal{U}(G, MN) +- \mathcal{U}(G, N) \subseteq \mathcal{U}(G, MN +- N).</math>{{sfn|Jarchow|1981|pp=43-55}}</li>

and similarly{{sfn|Jarchow|1981|pp=43-55}}

<li><math display=block>\mathcal{U}(G, NM) -+ \mathcal{U}(G, N) \subseteq \mathcal{U}(G, NM -+ N).</math>{{sfn|Narici|Beckenstein|2011|pp=19-45}}</li>

<li><math>\mathcal{U}(G \cup H, M \cap N) \subseteq \mathcal{U}(G, M) \cap \mathcal{U}(H, N)</math> forFor any subsets <math>G, H \subseteq X</math> and any non-empty subsets <math>M, N \subseteq Y.,</math>{{sfn|Jarchow|1981|pp=43-55}} For such subsets, it follows that,

<math display=block>\mathcal{U}(G \cup H, M \cap N) \subseteq \mathcal{U}(G, M) \cap \mathcal{U}(H, N)</math>

which implies:

<ul>

<li>if <math>M \subseteq N</math> then <math>\mathcal{U}(G, M) \subseteq \mathcal{U}(G, N).</math>{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>

Line 51 ⟶ 60:

<li>For any <math>M, N \in \mathcal{N}</math> and subsets <math>G, H, K</math> of <math>T,</math> if <math>G \cup H \subseteq K</math> then <math>\mathcal{U}(K, M \cap N) \subseteq \mathcal{U}(G, M) \cap \mathcal{U}(H, N).</math></li>

</ul>

</li>

<li>For any family <math>\mathcal{S}</math> of subsets of <math>T</math> and any family <math>\mathcal{M}</math> of neighborhoods of the origin in <math>Y,</math>{{sfn|Narici|Beckenstein|2011|pp=19-45}} <math display="block">\mathcal{U}\left(\bigcup_{S \in \mathcal{S}} S, N\right) = \bigcap_{S \in \mathcal{S}} \mathcal{U}(S, N) \qquad \text{ and } \qquad \mathcal{U}\left(G, \bigcap_{M \in \mathcal{M}} M\right) = \bigcap_{M \in \mathcal{M}} \mathcal{U}(G, M).</math></li>

<li><math>\mathcal{U}(\varnothing, N) = F.</math></li>

<li><math>\mathcal{U}(G, N) - \mathcal{U}(G, N) \subseteq \mathcal{U}(G, N - N).</math>{{sfn|Narici|Beckenstein|2011|pp=19-45}}</li>

<li><math>\mathcal{U}(G, M) + \mathcal{U}(G, N) \subseteq \mathcal{U}(G, M + N).</math>{{sfn|Jarchow|1981|pp=43-55}}</li>

<li>For any family <math>\mathcal{S}</math> of subsets of <math>T</math> and any family <math>\mathcal{M}</math> of neighborhoods of the origin in <math>Y,</math>{{sfn|Narici|Beckenstein|2011|pp=19-45}} <math display="block">\mathcal{U}\left(\bigcup_{S \in \mathcal{S}} S, N\right) = \bigcap_{S \in \mathcal{S}} \mathcal{U}(S, N) \qquad \text{ and } \qquad \mathcal{U}\left(G, \bigcap_{M \in \mathcal{M}} M\right) = \bigcap_{M \in \mathcal{M}} \mathcal{U}(G, M).</math></li>

</ul>

===Uniform structure===