Infinite dimentional subspace need not be closed

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SUMMARY

The discussion centers on the properties of the subspace Y, defined as the span of polynomials in the space X=C[0,1]. It is established that Y is not closed in X, contradicting a claim made by a participant who argued that Y is closed due to it being a subset of X. The conclusion drawn is that the infinite-dimensional subspace of polynomials does not contain all its limit points, confirming that Y is indeed not closed in X.

PREREQUISITES
  • Understanding of functional analysis concepts, specifically C[0,1]
  • Knowledge of polynomial functions and their properties
  • Familiarity with the concept of closure in topological spaces
  • Basic understanding of linear combinations and spans in vector spaces
NEXT STEPS
  • Study the properties of closed sets in functional analysis
  • Explore the concept of infinite-dimensional vector spaces
  • Learn about the topology of C[0,1] and its implications for subspaces
  • Investigate the relationship between polynomials and their convergence in functional spaces
USEFUL FOR

Mathematicians, students of functional analysis, and anyone interested in the properties of polynomial spaces and their topological characteristics.

nwl
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Let X=C[O,1] and Y=span($X_{0},X_{1},···$), where $X_{j}={t}^{i}$, so that Y is the set of all polynomials. Y is not closed in X.
 
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nwl said:
Let X=C[O,1] and Y=span(x\_{0},x\_{1},···), where X\_{j}={t}^{i}, so that Y is the set of all polynomials. Y is not closed in X.
Are you asking a question here or are you simply informing us? We can't help you if we don't know what you are asking about!

-Dan
 


I would have to disagree with your statement that Y is not closed in X. Y is indeed closed in X, as it is the span of all polynomials in X. This means that any linear combination of polynomials in X will also be a polynomial in X, and thus Y is a subset of X. Since Y is a subset of X and contains all polynomials in X, it is closed in X. Therefore, Y is closed in X.
 

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