I'm studying functional analysis, specifically metric space section using Kreyzig's Introductory Functional Analysis with Application textbook.
It explains sequence space $l^\infty$ as a set $X$ that take the set of all bounded sequences of complex numbers; that is, every element of $X$ is a bounded complex sequence. The metric is defined by $$d(x,y) = \sup_{i\in N}|x_i – y_i|$$
My question is what constitute of elements of set $X$ in sequence space $l^\infty$. I thought of 2 specific examples but I'm not sure which one is right.
Case 1:
$x \in X$ and $y \in X$ such that:
$$ x = ({1}/{n^2})$$
$$ y = (1,1,1,1,1)$$
in this case,
$$d(x,y) = \sup_{i\in N}|{1}/{1^2}-1, {1}/{2^2}-1, {1}/{3^2}-1,…|$$
Case 2:
Set $X$ is set of $(a_n)$ such that $$(a_n) = ({1}/{n^2})$$
$x \in X$ and $y \in X$ such that say
$$x = (1/1^2, 1/2^2, 1/3^2…)$$ and
$$y = (1/100^2, 1/101^2, 1/102^2…)$$
The difference between $x$ and $y$ in this case is they start indexing at different starting point.
In this case,
$$d(x,y) = \sup_{i\in N}|{1}/{1^2}-{1}/{100^2}, {1}/{2^2}-{1}/{101^2}, {1}/{3^2}-{1}/{102^2},…|$$
I been thinking about it but couldn't wrap my mind on which case is the book referring to. Wonder if someone could point it out to me.
Thank you!
Best Answer
To be clear, start a little further back:
A sequence in $\Bbb C$ is just a function $a : \Bbb N \to \Bbb C$.
For $n \in \Bbb N$, we commonly denote $a(n)$ by $a_n$. And often we denote $a$ itself by $\{a_n\}_{n=0}^\infty$ or just by $\{a_n\}$ for short. We can also think of $a$ as an infinite tuple: $a = (a_0, a_1, a_2, ...)$.
A sequence is bounded if there is a real number $M$ such that for all $n, |a_n| < M$.
So $\ell^\infty = \{a \mid a$ is a bounded sequence in $\Bbb C\}$.
Now lets look at your sequences:
(I have to be careful here since I define sequences as being for all of $\Bbb N$ and am further using the definition of $\Bbb N$ that includes $0$.)
(1) $x : \Bbb N \to \Bbb C : n \mapsto \begin{cases}0& n=0\\1/n^2 & n > 0\end{cases}$.
Since $|x_n| = x_n < 2$ for all $n$, $x_n$ is a bounded sequence. Therefore $x \in \ell^\infty$.
(2) $y : \Bbb N \to \Bbb C : n \mapsto 1$.
For all $n$, $|y_n| = 1 < 2001$. Therefore $y$ is a bounded sequence. So $y \in \ell^\infty$.
Finally, your third sequence (the 2nd $x$ is the same as the first) is
(3) $\displaystyle y : \Bbb N \to \Bbb C : n \mapsto \frac 1{(n + 100)^2}$.
Since $n + 100 > 1$, we have $1/(n+100)^2 < 1$ for all $n$. Thus $|y_n| = y_n < 1$. So $y$ is a bounded sequence and so $y \in \ell^\infty$.
So all the sequences you gave are just examples of sequences in $\ell^\infty$.
You talk about sets "$X$", but have not provided any understandable definition for them, so I cannot address what you meant by it.