What is an example of a non-closed subspace in a normed space?

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Discussion Overview

The discussion revolves around identifying examples of non-closed subspaces within normed spaces, specifically focusing on various function spaces and their properties. Participants explore theoretical aspects and provide specific examples to illustrate their points.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant asks for an example of a non-closed subspace in a normed space.
  • Another participant suggests the space of continuous functions on [0,1] with the supremum norm, stating that the subspace of differentiable functions is not closed.
  • A different participant mentions the linear span of a complete orthonormal set in Hilbert space, noting it is dense but not closed since not all vectors are finite linear combinations.
  • Another example provided is the set of indefinitely differentiable functions with compact support in L^p(R), which is dense and thus not closed.
  • One participant requests clarification on why the space of differentiable functions is not closed, prompting another to explain that a sequence of differentiable functions can converge uniformly to a non-differentiable function.
  • A participant introduces the space of integrable functions on [0,1] with the L_1 norm, highlighting a Cauchy sequence of continuous functions whose limit is integrable but discontinuous, thus not in the subspace of continuous functions.
  • Another participant requests a specific example, indicating a desire for further clarification.

Areas of Agreement / Disagreement

Participants present multiple examples of non-closed subspaces, indicating a lack of consensus on a singular example. The discussion remains unresolved as various viewpoints and examples are shared without a definitive conclusion.

Contextual Notes

Some examples depend on specific properties of function spaces, such as uniform convergence and the nature of differentiability, which may not be universally applicable across all normed spaces.

Who May Find This Useful

Mathematicians, students, and researchers interested in functional analysis, particularly those exploring properties of normed spaces and subspaces.

quasar987
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What would be an example of a not (topologically) closed subspace of a normed space?
 
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Consider the space of continuous functions f:[0,1]->R with the supremum norm
\Vert f\Vert=\sup |f(x)|. This is a normed vector space (in fact, a Banach space). The subspace of differentiable functions is not closed.

mathboy said:
R is a normed space, so take any open interval.

That's not a linear subspace though.
 
the linear span of a complete orthonormal set in hilbert space. it is dense, since all vectors are infinite series expansions of the, but not closed since not all vecors are finite linear combinations.

i.e. a hilbert basis is an o.n. set whose span is dense.
 
Or the set of indefinitely differentiable functions with compact support defined on R as a subset of L^p(R). It is a proper subspace and it is dense, therefor it is not closed.

I get the idea, thanks!
 
why the space of diffrental function not closed help me pleas quakly
 
lady99 said:
why the space of diffrental function not closed help me pleas quakly

Because you can find an example of a sequence of differentiable functions that converge uniformly to a non-differentiable function.
 
Simply take the space X of integrable functions on [0,1], equipped with the L_1 norm, and consider the subspace Y of continuous functions on [0,1]: one can find a Cauchy sequence of functions in Y whose limit is integrable but discontinuous, and is hence no longer in Y.
 
pleas give me eaxample
 

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