Missouri Journal of Mathematical Sciences

A Continuous Bijection from $\ell^{2}$ Onto a Subset of $\ell^{2}$ Whose Inverse is Everywhere Unboundedly Discontinuous, with an Application to Packing of Balls in $\ell^2$

Sam H. Creswell

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Abstract

There is a continuous bijection from $ \ell^{2}$ onto a subset of $\ell^{2}$ whose inverse is everywhere unboundedly discontinuous. If $B$ is a ball in $\ell^2$, then the continuous bijection defined on $\ell^2$ maps countably many mutually disjoint balls of $\ell^2$ into countably many mutually disjoint balls in $B$, making those images mutually disjoint.

Article information

Source
Missouri J. Math. Sci., Volume 23, Issue 1 (2011), 12-18.

Dates
First available in Project Euclid: 1 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.mjms/1312233179

Digital Object Identifier
doi:10.35834/mjms/1312233179

Mathematical Reviews number (MathSciNet)
MR2828729

Zentralblatt MATH identifier
1235.46031

Subjects
Primary: 46C05: Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
Secondary: 46B20: Geometry and structure of normed linear spaces

Citation

Creswell, Sam H. A Continuous Bijection from $\ell^{2}$ Onto a Subset of $\ell^{2}$ Whose Inverse is Everywhere Unboundedly Discontinuous, with an Application to Packing of Balls in $\ell^2$. Missouri J. Math. Sci. 23 (2011), no. 1, 12--18. doi:10.35834/mjms/1312233179. https://projecteuclid.org/euclid.mjms/1312233179


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References

  • S. H. Creswell, A continuous bijection from $\ell^{2}$ onto a subset of $\ell^{2}$ whose inverse is everywhere discontinuous, Amer. Math. Monthly, 117.9 (2010), 823–828.
  • S. H. Creswell, Uncountably many mutually disjoint, dense and convex subsets of $\ell^{2}$ with applications to path connected subsets of spheres, Missouri Journal of Mathematical Sciences, 21 (2009), 163–174.
  • N. Young, An Introduction to Hilbert Space, Cambridge University Press, Cambridge, 1988.