Is the set of bounded signals considered in topology and C^1 functions compact?

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Discussion Overview

The discussion revolves around the compactness of the set of bounded signals defined as X = { x: |x(t)| ≤ X_max, ∀ t }. Participants are exploring this concept within the context of topology and function spaces, particularly focusing on the implications of boundedness and the relevant topological structures.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the compactness of the set of bounded signals, seeking clarification on the conditions that would support or refute this claim.
  • Another participant asserts that boundedness alone does not imply compactness in the topology of uniform convergence.
  • A further inquiry is made regarding the specific topology and function space being considered in relation to the bounded signals.
  • There is speculation about the nature of the signals, with one participant suggesting they may be periodic functions that are almost everywhere C^1.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the compactness of the set of bounded signals, and multiple viewpoints regarding the topology and nature of the functions remain present.

Contextual Notes

The discussion lacks clarity on the specific topological space and the definitions being used, which may affect the conclusions about compactness.

symv
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I am considering the set of all bounded signals given by

[tex]X = \left\{ x:\ |x(t)| \leq X_{\max}, \forall t \right\}.[/tex]

Is this set compact? Can anyone help me?

Thank you guys
 
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symv said:
I am considering the set of all bounded signals given by

[tex]X = \left\{ x:\ |x(t)| \leq X_{\max}, \forall t \right\}.[/tex]

Is this set compact? Can anyone help me?

Thank you guys



No, bounded alone does not imply compact
(in the topology of uniform convergence).
 
in what topology, in what space of functions ?
 
g_edgar said:
in what topology, in what space of functions ?


He said signals, so I guess it's something like periodic functions
that are a.e. C^1.
 

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