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mathplease
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I would like a second opinion on my answer to this question as I'm confusing myself thinking about my proof. Any input is appreciated
"Let (X, ||.||) be a complete normed linear space and Y [tex]\subset[/tex]X be a non-empty subspace of X. Then (Y, ||.||) is a normed linear space. Show that Y is complete if and only if it is closed."
convergent sequence: http://mathworld.wolfram.com/ConvergentSequence.html"
cauchy sequence: http://mathworld.wolfram.com/CauchySequence.html"
complete: a normed linear space in which every cauchy seq is convergent is complete
closed: (X,||.||) is a normed linear space. A is closed if {xn} [tex]\subseteq[/tex] A [tex]\subseteq[/tex] X and xn-> x then x [tex]\in[/tex]A.
Let {yn} be a Cauchy sequence in Y. Since (X,||.||) is complete, yn converges to y[tex]\in[/tex]X. Assuming Y is closed: y[tex]\in[/tex]Y.
Hence, Y is complete.
Conversely,
assume Y is complete. Let {yn} be a convergent sequence in Y. Since convergent sequences are Cauchy, {yn} is a Cauchy sequence.
Since Y is a complete normed linear space yn[tex]\rightarrow[/tex]y [tex]\in[/tex]Y (Cauchy sequences converge).
Hence Y is closed.
Therefore Y is complete if and only if it is closed.
Homework Statement
"Let (X, ||.||) be a complete normed linear space and Y [tex]\subset[/tex]X be a non-empty subspace of X. Then (Y, ||.||) is a normed linear space. Show that Y is complete if and only if it is closed."
Homework Equations
convergent sequence: http://mathworld.wolfram.com/ConvergentSequence.html"
cauchy sequence: http://mathworld.wolfram.com/CauchySequence.html"
complete: a normed linear space in which every cauchy seq is convergent is complete
closed: (X,||.||) is a normed linear space. A is closed if {xn} [tex]\subseteq[/tex] A [tex]\subseteq[/tex] X and xn-> x then x [tex]\in[/tex]A.
The Attempt at a Solution
Let {yn} be a Cauchy sequence in Y. Since (X,||.||) is complete, yn converges to y[tex]\in[/tex]X. Assuming Y is closed: y[tex]\in[/tex]Y.
Hence, Y is complete.
Conversely,
assume Y is complete. Let {yn} be a convergent sequence in Y. Since convergent sequences are Cauchy, {yn} is a Cauchy sequence.
Since Y is a complete normed linear space yn[tex]\rightarrow[/tex]y [tex]\in[/tex]Y (Cauchy sequences converge).
Hence Y is closed.
Therefore Y is complete if and only if it is closed.
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