This diagram appears in the last chapter of my sophomore-level analysis textbook, and there is no explanation of the sequence space or function space in this diagram because it is beyond the scope of this book!
I'm already know first, second, fifth row works. It is range of my textbook.
So, I want to ask you. What does $A[-\pi,\pi], A(\mathbb Z), \mathfrak X$ means? What is their name? What is their properties?
$$ \require{AMScd} \begin{CD} \{\text{analytic periodic function}\}@>\text{Fourier}>>{\{a\mid\limsup_n|a(n)|^{\frac1n}<1\}}\\ @A{\cap}AA @A{\cap}AA\\ \{C^\infty-\text{periodic function}\}@>\text{Fourier}>>\{\text{rapidly decreasing sequence}\}\\ @A{\cap}AA @A{\cap}AA\\ A[-\pi,\pi]@>\text{Fourier}>>\ell^1(\mathbb Z)\\ @A{\cap}AA @A{\cap}AA\\ \{\text{continuous periodic function}\}@>\text{Fourier}>>\mathfrak X\\ @A{\cap}AA @A{\cap}AA\\ L^2[-\pi,\pi]@>\text{Fourier}>>\ell^2(\mathbb Z)\\ @A{\cap}AA @A{\cap}AA\\ L^1[-\pi,\pi]@>\text{Fourier}>>A(\mathbb Z)\\ @. @A{\cap}AA\\ @.c_0(\mathbb Z) \end{CD} $$
In Fourier analyisis it is usually denoted:
$$A[-\pi,\pi]=\lbrace f\in L^1[-\pi,\pi]| \sum_{n=-\infty}^{\infty}|\hat{f}(n)|<\infty\rbrace$$
On the other hand I've never seen the notation $\mathfrak{X}$, but the notation $\mathscr{S}$. This is the Schwarz space:
$$\mathscr{S}=\lbrace f\in C^\infty(\mathbb{R})| |x|^n f^{m)}(x)~\text{bounded}~\forall~n,m\in\mathbb{N}\rbrace$$
No idea what does $A(\mathbb{Z})$ means. I'm not a Fourier analysis expert, but hope this helps.