Please define $\ell^2$ space and give some examples of its subspaces

Question

I have googled alot about $\ell^2$ space and its subspace. There are showing result for $L^2$. Is (small $\ell$) $\ell^2$ space and (greater $L$) $L^2$ space are same?

Answer 1

No. Take $1 \leq p < +\infty$. For any measure space $(X,\mathcal{A},\mu)$ we have that $$L^p(X,\mathcal{A},\mu) = \left\{[f] \mid \int_X |f(x)|^p\,{\rm d}\mu(x) < +\infty \right\},$$where $[f]$ denotes the equivalence class of $f\colon X \to \Bbb C$ under the relation $f \sim g$ if $f = g$ $\mu$-almost everywhere. People get lazy and conflate $[f]$ with $f$, writing just $L^p(X,\mu)$ or $L^p(X)$ instead, when the $\sigma$-algebra $\mathcal{A}$ and the measure $\mu$ are understood.

On the other hand, if $I$ is any index set, one has $$\ell^p(I) = \left\{(x_i)_{i \in I} \mid x_i \in \Bbb C \mbox{ for all }i \in I \mbox{ and }\sum_{i \in I} |x_i|^p<+\infty \right\}. $$The most common cases are when $I = \Bbb N$ or $I = \Bbb Z$, so the sum above becomes a series or a two-sided series.

We have that $\ell^p(I) = L^p(I, \wp(I), {\rm c})$, where ${\rm c}$ is the counting measure on $I$, since a family $(x_i)_{i \in I}$ is nothing more than a function $x\colon I \to \Bbb C$, with us writing $x_i$ for the value $x(i)$. You can replace $\Bbb C$ everywhere by $\Bbb R$ to get real versions of everything above.