P-Sequence Space with P-Norm forms Banach Space

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Theorem

Let $\ell^p$ be a p-sequence space.

Let $\norm {\, \cdot \,}_p$ be a p-norm.


Then $\struct {\ell^p, \norm {\, \cdot \,}_p}$ is a Banach space.


Proof

A Banach space is a normed vector space, where a Cauchy sequence converges with respect to the supplied norm.

To prove the theorem, we need to show that a Cauchy sequence in $\struct {\ell^p, \norm {\,\cdot\,}_p}$ converges.

We take a Cauchy sequence $\sequence {x_n}_{n \mathop \in \N}$ in $\struct {\ell^p, \norm {\,\cdot\,}_p}$.

Then we consider the $k$th component and show, that a real Cauchy sequence $\sequence {x_n^{\paren k}}_{n \mathop \in \N}$ converges in $\struct {\R, \size {\, \cdot \,}}$ with the limit $x^{\paren k}$ and denote the entire set as $\mathbf x$.

Finally, we show that $\sequence {\mathbf x_n}_{n \in \N}$, composed of components $x_n^{\paren k},$ converges in $\struct {\ell^p, \norm {\,\cdot\,}_p}$ with the limit $\mathbf x$.


Let $\sequence {\mathbf x_n}_{n \mathop \in \N}$ be a Cauchy sequence in $\struct {\ell^p, \norm{\, \cdot \,}_p}$.

Denote the $k$th component of $\mathbf x_n$ by $x_n^{\paren k}$.


$\sequence {x_n^{\paren k}}_{n \mathop \in \N}$ converges in $\struct {\R, \size {\, \cdot \,}}$

Let $\epsilon >0$.

Then:

$\ds \exists N \in \N : \forall m,n \in \N : m,n > N : \norm {\mathbf x_n - \mathbf x_m}_p < \epsilon$

For same $N, m, n$ consider $\size {x_n^{\paren k} - x_m^{\paren k} } $:

\(\ds \size {x_n^{\paren k} - x_m^{\paren k} }\) \(=\) \(\ds \paren {\size {x_n^{\paren k} - x_m^{\paren k} }^p}^{\frac 1 p}\)
\(\ds \) \(\le\) \(\ds \paren {\sum_{k \mathop = 0}^\infty \size {x_n^{\paren k} - x_m^{\paren k} }^p}^{\frac 1 p}\)
\(\ds \) \(=\) \(\ds \norm {\mathbf x_n - \mathbf x_m}_p\) Definition of P-Norm
\(\ds \) \(<\) \(\ds \epsilon\)

Hence, $\sequence {x_n^{\paren k}}_{n \mathop \in \N}$ is a Cauchy sequence in $\struct {\R, \size {\, \cdot \,}}$.

From Real Number Line is Complete Metric Space, $\R$ is a complete metric space.

Consequently, $\sequence {x_n^{\paren k}}_{n \mathop \in \N}$ converges in $\struct {\R, \size {\, \cdot \,}}$.


$\Box$

Denote the limit $\ds \lim_{n \mathop \to \infty} \sequence {x_n^{\paren k}}_{n \mathop \in \N} = x^{\paren k}$.

Denote $\sequence {x^{\paren k}}_{k \mathop \in \N} = \mathbf x$.


$\mathbf x$ belongs to $\ell^p$

Let $\epsilon > 0$.

Then:

$\exists N \in \N : \forall n,m \in \N : n,m > N : \norm {\mathbf x_n - \mathbf x_m}_p < \epsilon$.

Let $K \in \N$, $1 \le p < \infty$.

Then:

\(\ds \sum_{k \mathop = 1}^K \size {x_n^{\paren k} - x_m^{\paren k} }^p\) \(\le\) \(\ds \sum_{k \mathop = 1}^\infty \size {x_n^{\paren k} - x_m^{\paren k} }^p\)
\(\ds \) \(=\) \(\ds \norm {\mathbf x_n - \mathbf x_m}_p^p\)
\(\ds \) \(<\) \(\ds \epsilon^p\)
\(\ds \) \(<\) \(\ds \infty\)

Take the limit $m \to \infty$:

$\ds \sum_{k \mathop = 1}^K \size {x_n^{\paren k} - x^{\paren k}}^p < \ds \epsilon^p < \infty$

Since $K$ was arbitrary, we can take the limit $K \to \infty$.

By definition, $\forall k \in \N : x_n^{\paren k} - x^{\paren k} \in \R$.

In other words, $\mathbf x_n - \mathbf x \in \ell^p$.

But, by assumption, $\mathbf x_n \in \ell^p$.

Therefore:

$\paren {\mathbf x - \mathbf x_n} + \mathbf x_n = \mathbf x \in \ell^p$

$\Box$


$\sequence {\mathbf x_n}_{n \mathop \in \N}$ converges in $\struct {\ell^p, \norm {\, \cdot \,}_p}$ to $\mathbf x$

Let $1 \le p < \infty$.

Let $\epsilon > 0$.

Fix $N \in \N$, $m > N$.

Then:

$\ds \forall n > N : \norm {\mathbf x_n - \mathbf x_m}_p < \epsilon$

We have that $\sequence {x_n^{\paren k}}_{n \mathop \in \N}$ converges in $\struct {\R, \size {\, \cdot \,}}$.

Take the limit $m \to \infty$:

\(\ds \lim_{m \mathop \to \infty} \norm {\mathbf x_n - \mathbf x_m}_p\) \(=\) \(\ds \lim_{m \mathop \to \infty} \paren {\sum_{k \mathop = 0}^\infty \size {x_n^{\paren k} - x_m^{\paren k} }^p }^{\frac 1 p}\)
\(\ds \) \(=\) \(\ds \paren {\sum_{k \mathop = 0}^\infty \size {x_n^{\paren k} - x^{\paren k} }^p }^{\frac 1 p}\) Limit of Composite Function
\(\ds \) \(=\) \(\ds \norm {\mathbf x_n - \mathbf x}_p\)
\(\ds \) \(<\) \(\ds \epsilon\)

So $\sequence {\mathbf x_n}_{n \mathop \in \N}$ converges in $\struct {\ell^p, \norm {\, \cdot \,}_p}$.

$\blacksquare$


Also see


Sources

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