Proving Space of Differential Functions Not Closed

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Discussion Overview

The discussion revolves around the question of proving that the space of differentiable functions is not closed. Participants seek clarification on the context, specifically whether "closed" refers to topological closure, closure in a vector space, or another concept, and what specific types of functions or spaces are being considered.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant asks for clarification on whether "closed" refers to topology or as a subspace of a vector space.
  • Another participant suggests that the original poster might be interested in the closure of differentiable functions between two topological spaces, but seeks confirmation on which topology is being referenced.
  • A participant interprets "c[a, b]" as the set of continuous functions on the interval [a, b] but notes a discrepancy with the term "differentiable," suggesting it should refer to "c1[a, b]."
  • There is a proposal that the goal may be to show that the set of all polynomials is not closed as a subset of all differentiable functions on some interval, but further clarification on the operation or topology is needed.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the specifics of the question, with multiple interpretations of "closed" and the types of functions involved. The discussion remains unresolved as participants seek more information.

Contextual Notes

There are limitations in the discussion due to missing definitions of the topology or operations being referenced, as well as the specific nature of the functions under consideration.

lady99
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how i prove that space of defrential function not closed?
 
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You need to give us more details. Closed in the sense of topology or closed in the sense of being a subspace of a vector space (or a group or whatever)? Do you mean differential? What sort of objects is it acting on? Just real-valued functions of one real variable?
 
{x:xis polynomail} subset c[a,b]}
 
Still not enough information! If I were to guess, then it would be that you want to know how to prove that the space of differentiable functions between 2 topological spaces is not closed under some topology.

Is this correct? If so, which topology? Don't hold back!
 
I would interpret "c[a, b]" as the set of functions continuous on [a, b] but the orignal post said "differentiable" which would be "c1[a, b]".

So I take it you want to show that the set of all polynomials is not closed as a subset of the set of all differentiable functions on some interval.

But it is still not clear if you mean "closed" under some operation or "closed" in some topology. And, we would need to know what operation or topology is involved.
 

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