Notations in Functional Analysis

Question

I'm approaching the study of metric spaces, functional analysis, Hilbert spaces and so on and I have some questions in order to understand if I got it right.

For example, the space $\ell^1([0, 1])$: this is a Banach space, the space of the sequences which obey to $\sum_n \vert x_n\vert < +\infty$. Does the $[0, 1]$ part mean that the $x_n$ terms can assume values only between $0$ and $1$? Did I understand well?

Also I would like some clarification about this: "a sequence in $\ell^1$ is a sequence of sequences". Can you please provide me some example?

I'm trying to understand it correctly. So for example, let's take the sequences

$$a_n = \dfrac{1}{\log(n+2)} \qquad \qquad \qquad b_n = \dfrac{1}{e^n}$$

Then $$a_n = \left(\dfrac{1}{\ln(2)}, \dfrac{1}{\ln(3)}, \dfrac{1}{\ln(4)}, \ldots \right)$$

$$b_n = \left(1, e^{-1}, e^{-2}, \ldots\right)$$

I can hypothetically say they are in $\ell^1([0, 1])$ for each $x_n$ term assumes values between $0$ and $1$. Or maybe I understood nothing and I'm speaking malarkey...

(I meant hypothetically, because then I have to check if their sums converge too).

Please explain me where I am wrong, I tried to find some resounce online too but I cannot compare notes with some true human explanation.

Thank you!

Answer 1

For example, the space $\ell^1([0, 1])$: this is a Banach space, the space of the sequences which obey to $\sum_n \vert x_n\vert < +\infty$. Does the $[0, 1]$ part mean that the $x_n$ terms can assume values only between $0$ and $1$? Did I understand well?

That is a notation I've come to understand, yes. I've also, however, seen use of another notation where

$$\ell^p(I) := \left\{ (x_i)_{i \in I} \in \mathbb{C}^I \, \middle| \, \sum_{i \in I} |x_i|^p < \infty \right\}$$

(or another field of your choice). I suppose it depends on what you wish to leave implied for your definition: the field in question, or the indexing set.

Also I would like some clarification about this: "a sequence in $\ell^1$ is a sequence of sequences". Can you please provide me some example?

A sequence in a space is a countable subset thereof, given some order, loosely speaking. For instance, in $\mathbb{R}$, we have

$$ \left( \frac{1}{2^n} \right)_{n \in \mathbb{N}} = \left( \frac 1 2, \frac 1 4 , \frac 1 8 , \cdots \right)$$

as a sequence: each number is a real number. We may define a sequence of sequences as so: say we have

$$\left( x_n \right)_{n \in \mathbb{N}} \text{ where } x_n = \left( \xi_k^{(n)} \right)_{k \in \mathbb{N}}$$

For each $n$, we have a sequence $x_n$ (indexed by $n$), and each of these is a sequence itself indexed by $k$ (the $(n)$ being used to keep track of what members belong to a given sequence). For instance,

$$\left( x_n \right)_{n \in \mathbb{N}} \text{ where } x_n = \left( \frac n k \right)_{k \in \mathbb{N}}$$

Then we have

$$\begin{align*} x_1 &= \left( \frac 1 1 , \frac 1 2 , \frac 1 3 , \frac 1 4 , \cdots \right) \\ x_2 &= \left( \frac 2 1 , \frac 2 2 , \frac 2 3 , \frac 2 4 , \cdots \right) \\ x_3 &= \left( \frac 3 1 , \frac 3 2 , \frac 3 3 , \frac 3 4 , \cdots \right) \\ x_4 &= \left( \frac 4 1 , \frac 4 2 , \frac 4 3 , \frac 4 4 , \cdots \right) \end{align*}$$

(This is more focused on the "sequence of sequences" notion than $\ell^p$, mind you; each of these sequences are in $\ell^p$ for $p>1$ though. Again, notice that each $x_n$ is itself a separate element of the set of all sequences: each $x_n$ is itself a sequence, and we built a sequence of them.)

$$a_n = \left(\dfrac{1}{\ln(2)}, \dfrac{1}{\ln(3)}, \dfrac{1}{\ln(4)}, \ldots \right)$$

This sequence does not lie in $\ell^1$, owing to the summation condition. This is because

$$\ln(n) < n$$

for $n$ sufficiently large, so eventually

$$\frac{1}{\ln(n)} > \frac 1 n \text{ but } \sum_{n=1}^\infty \frac 1 n \text{ diverges}$$