Difference between spaces $\ell^2$, $L^2$, and $H^2$ Can you please help to understand what exactly the definitions of these spaces, actually I'm so confused.
Firstly, the Hilbert space $\ell^2 =\{ \{a_n\}_0^\infty: \sum_0^\infty |a_n|^2 <\infty$ }. Then can I conclude that if $(a_0, a_1, a_2, ...) \in \ell^2 \implies a_0, a_1, a_2, ... \in \mathbb{D}$ $(\mathbb{D}=\{ z \in \mathbb{C}: |z|<1 \})$??
Secondly, the Hilbert space $L^2 = \{ f \in S^1: \frac{1}{2\pi} \int_0^{2\pi} |f|^2 < \infty \}$  (all lebesgue integrable functions with finite square-integral), where $S^1 = \{ z \in \mathbb{C}: |z|=1 \}$. Did it implies that every analytic function $f : S^1 \to S^1$ is in $L^2$,? How can I write a function $f \in L^2$ ?
Thirdly, the Hardy-Hilbert space $ H^2 = \{ f: f(z) = \sum_0 ^ \infty a_n z^n \; and \; \sum_0^\infty |a_n|^2 < \infty \} $. Can I conclude that every function $f \in H^2 \implies f$ is analytic and $f: \mathbb{D} \to \mathbb{D}$ ??
Is true that $\ell^2 \subset H^2 \subset L^2$?? If does, how can it be?
I'm so sorry for this long question, but I'm really confused.
 A: *

**can I conclude that... * No, why should $|a_n| < 1$?  This being a vector space, every scalar multiple of a member of the space is also in the space.  So the possible values can't be bounded.


*No, not $f \in \mathbb S^1$, rather complex-valued measurable functions defined on $\mathbb S^1$.
2a) Did it implies that... By "analytic function $f: S^1 \to S^1$ I suppose you mean
a function analytic in a neighbourhood of $S^1$, and whose values on $S^1$ happen to fall in $S^1$.  It's a nice complex-analysis exercise to characterize what these
are: there are probably fewer than you realize.  Anyway, since the values are in $S^1$ they are bounded and they are certainly measurable, so yes they are in $L^2$.
2b) How can I write a function $f \in L^2$? I don't understand the question.  Write it as you would write any function.


*Can I conclude that... Since the power series converges absolutely for $|z| < 1$, it is analytic on $\mathbb D$, but again there's no reason to think the values are in $\mathbb D$.


*Is it true that $\ell^2 \subset H^2 \subset L^2$?  They are different sorts of things.  A member of $\ell^2$ is a sequence, not a function.
A member of $H^2$ is a function on $\mathbb D$, while a member of $L^2$ is a function on $\mathbb S^1$.  However, a function $f$ in $H^2$ does have radial limits almost everywhere on $\mathbb S^1$; the map taking $f$ to the function $f^*$ given by
$f^*(s) = \lim_{r \to 1-} f(rs)$ turns out to be an isometry from $H^2$ to
a subspace of $L^2(\mathbb S^1)$.  In other words, $H^2$ can be identified
in a natural way as a subspace of $L^2$.
