To show the sequence space $l_p$ is Banach space.

The $l_p$ space is already equipped with the $l_p$-norm, defined as:

\begin{align*} \|x\|_p = \left(\sum_{i \in \mathbb{N}} |x_i|^p\right)^{\frac{1}{p}} \qquad \text{for each} \; x=(x_i)_{1}^{\infty} \in X=l_p, \; 1\le p \lt \infty \end{align*}

thus we shall just show it is complete in order to be Banach.

I see a proof in a textbook but I DON'T understand. It begins as follows:

Let $(x_n)_1^{\infty}$ be a Cauchy sequence in $l_p$. We shall denote each member of this sequence by \begin{align*} x_n = \left(x_n(1),x_n(2),...\right) \end{align*}

Q1: The $(x_n)_1^{\infty}$ stands for a sequence $x_n$ where $n=1,2,3,...\to \infty$, or a sequence $x_n^i$ where $n$ is just like the nickname of the sequence but $i=1,2,3,...\to \infty$?

Q2: If $n=1,2,3,...\to \infty$, then each member of this sequence should be $x_1,x_2,x_3,...$ and $(x_n)_1^{\infty} = (x_1,x_2,...)$, but why $x_n = \left(x_n(1),x_n(2),...\right)$?

Q3: If $(1),(2),...$ plays like a coordinate index, then $x_n(1),x_n(2),...$ should be components of $x_n$, but not "members of this sequence". I think the members of this sequence should be $x_1,x_2,...,x_m,x_n,...$

Q4: For example, $x_n=\frac{1}{n}$ for which $(x_n)_1^{\infty}$ is Cauchy, but $x_n(1)$ is meaningless, as I think there is no component or coordinate of $x_n=\frac{1}{n}$.

Then, given $\epsilon > 0$, there exists an $N(\epsilon) = N \in \mathbb{N}$ such that \begin{align*} \|x_n - x_m\|_p = \left(\sum_{i=1}^{\infty}|x_n(i) - x_m(i)|^p\right)^{\frac{1}{p}} < \epsilon \qquad \text{for all} \; m,n \ge N \end{align*}

Q5: By definition of the $l_p$-norm, the sum should be taken over the member of the sequence, namely the subscript $n$, i.e., $\|x\|_p = \left(\sum_{n\in\mathbb{N}}|x_n|^p\right)^{\frac{1}{p}}$ for $x=(x_n)_{1}^{\infty} \in l_p$. But in the proof the sum takes over the index $i$ rather than $n$ or $m$. Why?

Q6: For the Cauchy sequence $x = (x_n)_1^{\infty} = (x_1,x_2,x_3,...)$, in my opinion, $\|x\|_p = \left(|x_1|^p + |x_2|^p +|x_3|^p + ... \right)^{\frac{1}{p}} = \left(\sum_{n \in \mathbb{N}} |x_n|^p\right)^{\frac{1}{p}}$. In such a case I don't understand the definition of $\|x_n - x_m\|_p$.

Q7: If $x = (x_n)_1^{\infty} = (x_n^1,x_n^2,x_n^3,...)$ is the Cauchy sequence, and $x = (x_m)_1^{\infty} = (x_m^1,x_m^2,x_m^3,...)$ is another Cauchy sequence, then the members of each Cauchy sequence become closer and closer, i.e., $\|x_n^i-x_n^j\|_p \to 0$ as $i,j \to \infty$, BUT how can we say the distance between the two independent Cauchy sequences satisfies $\|x_n - x_m\|_p \to 0$ as $n,m \to \infty$?

I am totally confused about $(x_n)_1^{\infty} \in l_p$ here! I just started learning analysis. Could you please answer my Q1-Q7 and show me a particular example the $(x_n)_1^{\infty}$ looks like?


A1: The sequence $(x_n)_n$ is of elements in $\ell^p$. This means that each element $x_n$ is in $\ell^p$, so each element is itself a sequence. Instead of writing $$ x_n = ((x_n)_1, (x_n)_2, (x_n)_3, \ldots), $$ the author chose to use $$x_n = (x_n(1), x_n(2), x_n(3), \ldots)$$ to represent each element of the sequence. You can think of this notation as, instead of considering $\ell^p$ to be a special space of sequences, we see it as a special space of functions on $\mathbb{N}$. (These two are the same, really, since a sequence is just a function on $\mathbb{N}$.) From this point of view, each element in $y\in\ell^p$ is really a function $y:\mathbb{N}\to\mathbb{R}$ satisfying $\|y\|_p := \left(\sum_{n=1}^\infty |y(n)|^p\right)^{1/p} < \infty$. Thus $(x_n)_n$ is really a sequence of functions instead of a sequence of sequences.

A2: This is addressed in A1, but I'll reiterate. You have your sequence in $\ell^p$, which is $(x_n)_n = (x_1,x_2,\ldots)$. Each term of this sequence is in $\ell^p$, which means it is a sequence of real numbers (or function from $\mathbb{N}$ into $\mathbb{R}$): $x_n = (x_n(1), x_n(2), \ldots)$. So you have a sequence of sequences (or a sequence of functions).

A3: Again, there are many sequences at work here. The elements $x_1,x_2,\ldots$ in $\ell^p$ are components of the sequence $(x_n)_n$, and the elements $x_n(1),x_n(2),\ldots$ in $\mathbb{R}$ are components of the sequence $x_n$.

The questions Q4-Q7 seem to be stemming from the same confusion, so I'll stop here and give an example.

Consider $p=1$. We know from calculus that the sequences $(1/k^2)_k$, $(1/k^3)_k$, $(1/k^4)_k$, and so forth are elements of $\ell^1$. For each $n\in\mathbb{N}$, define $x_n:\mathbb{N}\to\mathbb{R}$ by $$ x_n(k) = \frac{1}{k^{n+1}}. $$ We will write the function $x_n$ as a sequence $(x_n(1), x_n(2), \ldots)$. Now note that $x_1=(1/k^2)_k$, $x_2=(1/k^3)_k$, and so forth. Thus each $x_n$ is in $\ell^1$ and so the sequence $(x_n)_n$ is a sequence of elements in $\ell^1$.


In short, in think, the author used it to denote the ordered array of "sequences of sequences". Each of the sequences denotes the component of a coordinate system. Taking them as individual real number lines, we apply the cauchy sequences to define completeness.

PS: I am just a newbie in analysis.


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