Difference between revisions of "Notes:Ell^p(C) is complete for p between one and positive infinity inclusive"
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* {{M|\forall x\in X\exists C\in\mathbb{R}_{>0}\forall a\in A[d(a,x)\le C]}} | * {{M|\forall x\in X\exists C\in\mathbb{R}_{>0}\forall a\in A[d(a,x)\le C]}} | ||
===Proof body=== | ===Proof body=== | ||
+ | * Let {{M|p\in[1,+\infty]\subseteq\overline{\mathbb{R} } }} be given. | ||
+ | ** Let {{M|(\mathbf{x}_n)_{n\in\mathbb{N} }\subseteq\ell^p(\mathbb{C})}} be a [[Cauchy sequence]] in {{M|\ell^p(\mathbb{C})}} | ||
+ | *** By lemma 1, we obtain, for each {{M|k\in\mathbb{N} }} the sequence: {{M|(x_m^k)_{m\in\mathbb{N} } }} and we know this is [[Cauchy sequence|Cauchy]] | ||
+ | *** As {{M|\mathbb{C} }} is a [[complete metric space]] - {{XXX|link to this}} - we see that {{M|(x_m^k)_m}} {{link|converges|sequence}} | ||
+ | **** For {{M|k\in\mathbb{N} }} define: {{MM|t_k:\eq\lim_{m\rightarrow\infty}(x^k_m)}} which we have just established exists | ||
+ | ***** Define {{M|\mathbf{t}:\eq(t_1,\ldots,t_k,\ldots)}} - a [[sequence]] | ||
+ | ****** We claim that: | ||
+ | ******# {{M|t\in\ell^p(\mathbb{C})}} | ||
+ | ******# {{MM|\lim_{n\rightarrow\infty}(\mathbf{x}_n)\eq\mathbf{t} }} |
Revision as of 18:41, 22 February 2017
Statement
[ilmath]\forall p\in[1,+\infty]\subseteq\overline{\mathbb{R} } [/ilmath] the space [ilmath]\ell^p(\mathbb{C})[/ilmath] is complete.
Proof
Leammas
Lemma 1:
- Let [ilmath](\mathbf{x}_n)_{n\in\mathbb{N} }\subseteq\ell^p(\mathbb{C})[/ilmath] be given
- Suppose [ilmath](\mathbf{x}_n)_{n\in\mathbb{N} } [/ilmath] is a Cauchy sequence
- Write them out as follows:
- [ilmath]\begin{array}{ccccc} \mathbf{x}_1 & \eq & (x_1^m)_{m\in\mathbb{N} } & \eq & x_1^1,\ x_1^2,\ \ldots ,\ x_1^k,\ \ldots \\ \mathbf{x}_2 & \eq & (x_2^m)_{m\in\mathbb{N} } & \eq & x_2^1,\ x_2^2,\ \ldots,\ x_2^k,\ \ldots \\ \vdots \end{array}[/ilmath]
- Now consider the [ilmath]k^\text{th} [/ilmath] column of this table, this gives us the sequence: [ilmath](x_m^k)_{m\in\mathbb{N} } [/ilmath]
- We claim that [ilmath](x_m^k)_{m\in\mathbb{N} } [/ilmath] is a Cauchy sequence itself
- Write them out as follows:
- Suppose [ilmath](\mathbf{x}_n)_{n\in\mathbb{N} } [/ilmath] is a Cauchy sequence
Lemma 2:
Let [ilmath](X,d)[/ilmath] be a metric space, then [ilmath]A\in\mathcal{P}(X)[/ilmath] is bounded in [ilmath]X[/ilmath] if and only if:
- [ilmath]\forall x\in X\exists C\in\mathbb{R}_{>0}\forall a\in A[d(a,x)\le C][/ilmath]
Proof body
- Let [ilmath]p\in[1,+\infty]\subseteq\overline{\mathbb{R} } [/ilmath] be given.
- Let [ilmath](\mathbf{x}_n)_{n\in\mathbb{N} }\subseteq\ell^p(\mathbb{C})[/ilmath] be a Cauchy sequence in [ilmath]\ell^p(\mathbb{C})[/ilmath]
- By lemma 1, we obtain, for each [ilmath]k\in\mathbb{N} [/ilmath] the sequence: [ilmath](x_m^k)_{m\in\mathbb{N} } [/ilmath] and we know this is Cauchy
- As [ilmath]\mathbb{C} [/ilmath] is a complete metric space - TODO: link to this- we see that [ilmath](x_m^k)_m[/ilmath] converges
- For [ilmath]k\in\mathbb{N} [/ilmath] define: [math]t_k:\eq\lim_{m\rightarrow\infty}(x^k_m)[/math] which we have just established exists
- Define [ilmath]\mathbf{t}:\eq(t_1,\ldots,t_k,\ldots)[/ilmath] - a sequence
- We claim that:
- [ilmath]t\in\ell^p(\mathbb{C})[/ilmath]
- [math]\lim_{n\rightarrow\infty}(\mathbf{x}_n)\eq\mathbf{t} [/math]
- We claim that:
- Define [ilmath]\mathbf{t}:\eq(t_1,\ldots,t_k,\ldots)[/ilmath] - a sequence
- For [ilmath]k\in\mathbb{N} [/ilmath] define: [math]t_k:\eq\lim_{m\rightarrow\infty}(x^k_m)[/math] which we have just established exists
- Let [ilmath](\mathbf{x}_n)_{n\in\mathbb{N} }\subseteq\ell^p(\mathbb{C})[/ilmath] be a Cauchy sequence in [ilmath]\ell^p(\mathbb{C})[/ilmath]