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Finite dimensional linear submanifold of N is complete.

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Suppose that N is a normed linear space. Prove that each finite dimensional linear submanifold of N is complete and therefore closed.

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Solution Summary

This solution is comprised of a detailed explanation of the problem on a normed linear space.
It contains step-by-step explanation for the following problem:
Normed Linear Space:

Suppose that N is a normed linear space. Prove that each finite dimensional linear submanifold
of N is complete and therefore closed.

Solution contains detailed step-by-step explanation.

Solution Preview

Please see the attached file.

Suppose that is a normed linear space.
Let be a finite dimensional linear submanifold of .
Hence itself is a finite dimensional normed linear space.

Now we have to prove that is complete and therefore closed.
Let B = be a basis for so that
-------------------------------------(1)
...

Solution provided by:
Thokchom Sarojkumar Sinha, MSc
Education
  • BSc, Manipur University
  • MSc, Kanpur University
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