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Proof Convergent Sequence Obvious Absolute Convergence Series Subspace Topology Topological Space Banach Space Convergents To A Continued Fraction Quotient Norm

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Quotients Of Banach Spaces By Closed Subspaces Are Banach Spaces Under The Quotient Norm

Theorem - Let be a Banach space and a closed subspace. Then with the quotient norm is a Banach space.

Proof : In order to prove that is a Banach space it is enough to prove that every series in that converges absolutely also converges in .

Let $\sum_{n} X_n$ be an absolutely convergent series in , i.e., $\sum_{n} \Vert X_n\Vert _{X/M} < \infty$ . By definition of the quotient norm, there exists $x_n \in X_n$ such that

$\displaystyle \Vert x_n \Vert \le \Vert X_n \Vert _{X/M} + 2^{-n}$

It is clear that $\sum_{n} \Vert x_n\Vert < \infty$ and so, as is a Banach space, $\sum_{n} x_n$ is convergent.

Let $x = \sum_{n} x_n$ and $s_k = \sum_{n=1}^k x_n$ . We have that

$\displaystyle x - s_k + M = (x+M) - (s_k +M) = (x+M) - \sum_{n=1}^k (x_n +M) = (x+M) - \sum_{n=1}^k X_n$

Since $\Vert x-s_k + M \Vert _{X/M} \leq \Vert x-s_k\Vert \longrightarrow 0$ we see that $\sum_n X_n$ converges in to . $\square$

1	Subspace Topology
1	Topological Space
1	Obvious
1	Proof
1	Absolute Convergence
1	Series
1	Convergent Sequence
2	Banach Space
5	Convergents To A Continued Fraction
12	Quotient Norm