Bounded inverse theorem

The completion of X is the space ${\displaystyle c_{0}}$ of all sequences that converge to zero, which is a (closed) subspace of the ℓ_p space ℓ_∞(N), which is the space of all bounded sequences. However, in this case, the map T is not onto, and thus not a bijection. To see this, one need simply note that the sequence

${\displaystyle x=\left(1,{\frac {1}{2}},{\frac {1}{3}},\dots \right),}$

is an element of ${\displaystyle c_{0}}$, but is not in the range of ${\displaystyle T:c_{0}\to c_{0}}$.

See also

• Almost open linear map
   – Map that satisfies a condition similar to that of being an open map.Pages displaying short descriptions of redirect targets
• Closed graph
   – Graph of a map closed in the product spacePages displaying short descriptions of redirect targets
• Closed graph theorem – Theorem relating continuity to graphs
• Open mapping theorem (functional analysis) – Condition for a linear operator to be open
• Surjection of Fréchet spaces – Characterization of surjectivity
• Webbed space – Space where open mapping and closed graph theorems hold

References

Bibliography

• OCLC 840293704
  .
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
  OCLC 144216834
  .
• Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. pp. 356.
  ISBN 0-387-00444-0
  . (Section 8.2)
• OCLC 849801114
  .