Does the finite sequence belong to $\ell^p$?

For $1\leqq p<\infty,$ $\ell^p$ is defined by $\ell^p=\{ \{x_n\}_{n=1}^\infty \subset K \mid \sum_{n=1}^\infty |x_n|^p<\infty\}.$

Does the arbitrary sequence $\{a_n\}_{n=1}^N \subset K$ belong to $\ell^p$ ?

Of course, $\sum_{n=1}^N |a_n|^p<\infty$ but I wonder whether $\{a_n \}_{n=1}^N$ belongs to $\ell^p$ because this is not the form $\{ \cdot \}_{n=1}^\infty$ but the form $\{ \cdot \}_{n=1}^N.$

I think that for given $\{a_n \}_{n=1}^N$, if I define $a_n=0$ for $n\geqq N+1$, I can check $\{a_n \}_{n=1}^\infty \in \ell^p$, but can I say $\{a_n \}_{n=1}^N$ itself is in $\ell^p$ ?

Or, in the first place, isn't $\{a_n\}_{n=1}^N$ called "sequence" ?

Solution 1:

My answer will have to be "yes and no".

In the formal sense, the answer is no. $\ell^p$ consists of elements of the form $a: \mathbb N \longrightarrow \mathbb C$. However, such a finite sequence is something of the form $a: \{1, \dots, N\} \longrightarrow \mathbb C$. This is not a function with domain $\mathbb N$ so strictly speaking it is not in $\ell^p$.

However, I think this question relates to a deeper concept than mere formalism - it's about how math is communicated. We are not always so strict in math. Surely, speaking strictly is useful. We must be precise when doing math, as subtle errors abound, especially when one is encountering a new topic. However, we often abuse notation and speak more loosely when we are working practically, as strictness can become cumbersome. Ultimately, math is a communicative process. What you write down is meant to be understood by another person doing math. If I read "$\{a_n\}_{n=1}^N \in \ell^p$" it would be immediately clear to me that what you meant is the extension by $0$ you describe. This is a routine thing to do, and contextually the meaning is clear. It may even be annoying for a reader if you were to belabor this point, but this depends on your audience. As with any form of speech, the particular language mathematicians use has imprecisions that are filled in by contextual understanding. It's the duty of a good author to determine what things their audience can pick up from context and what things must be spelled out.

Let me give another example. Is $\mathbb R \subseteq \mathbb C$? Well sure, we say this all the time. But often we consider $\mathbb C$ to be the set $\mathbb R^2$ with additional structure. Elements of $\mathbb R^2$ are pairs of real numbers, and a real number is not a pair of real numbers, so if we again insist on speaking strictly, then it's not true that $\mathbb R \subseteq \mathbb C$. But this is pedantry. We do freely think about the real numbers as sitting in the complex numbers without caring that on a technical set theoretic level it may not be strictly true. On that note, objects are not always even well defined. Are complex numbers pairs of real numbers? Cosets in $\mathbb R[x]/(x^2 + 1)$? Are real numbers equivalence classes of Cauchy sequences? Dedekind cuts? None of these sets contain one another, but does it really matter? When I say $\mathbb R \subseteq \mathbb C$ or $\{a_n\}_{n=1}^N \in \ell^p$, I expect my audience to be able to infer the meaning of these statements without my having to spell out all the little details. Doing so would be tedious for both me as an author, and for the reader to actually read.

I'll leave you with some rough guidelines, but I myself am a novice in mathematical writing. I'd encourage you to seek out experts who have written about this, such as Terry Tao on his blog (EDIT: This particular post on "compilation errors" is especially relevant) or any personal mentors you have. Additionally, having peers and members of your intended audience comment on what you've written is deeply important.

First and foremost, especially as a beginner, is to be correct and precise. You must be completely sure that everything you write down is correct, and that any skipped details are things you can readily fill in if asked. If you write down $\mathbb R \subseteq \mathbb C$ or $\{a_n\}_{n=1}^N \in \ell^p$, be prepared to defend it!

Secondly, think about who your audience is. Are you writing homework for a class? Perhaps then the grader will want to see every detail spelled out to determine if you are indeed capable of doing so. Are you preparing lecture notes for people learning analysis for the first time? Maybe this abuse of notation would confuse them. For advanced graduate students? They're probably used to things like this. Are these notes just for you? Are they for a textbook? Each audience and medium will have different expectations and standards. You should consider what your audience is likely to know, how much time they have to sit and digest the material itself, and what constraints the medium imposes - like if you have only an hour for a talk.