If $A\subset B,$ we may write $A\hookrightarrow B$ (According to this notion of Inclusion map).
My Questions: (1)If we have the notation $\subset$ (subset), then why do we need the notation $\hookrightarrow $ (Embedding)? How these two notations differs? If $A\hookrightarrow B$, then can we say $A\subset B$? (example?) (2) Sometimes authors says that the imbedding $A\hookrightarrow B$ is compact? What does this mean? (3) Is $\ell^{1}(\mathbb Z) \hookrightarrow \ell^{\infty}(\mathbb Z)$ is a compact embedding?
Given a subset $A \subset B$, the inclusion map $i : A \hookrightarrow B$ is an embedding.
Also, given a general embedding $f : A \hookrightarrow B$, its image $f(A)$ is a subset of $B$. However, $A$ itself in general is not a subset of $B$, and it should not be "thought of" as a subset of $B$ (despite what wikipedia says).
For example, the interval $[0,1] \subset \mathbb{R}$ may be embedded (topologically) into $\mathbb{R}^2$ by the function $$f(t) = (3t-7,-6t+22) $$ and its image is the line segment $L=\overline{PQ} \subset \mathbb{R}^2$ with endpoints $P=(-7,22)$ and $Q=(-4,16)$. However, that does not mean that we should think of $[0,1]$ and $L$ as "the same thing"; they are simply "isomorphic things".