Which normed space have Fatou's property? There is tool in mathematics, more specifically  in Analysis, which has immense applications in handling delicates estimates, limiting arguments, etc... ; namely Fatou property.
Let $E$ be a normed space, continuously embedded into $\mathcal{S'}(\mathbb R)$(= The space of tempered distributions). We say that $E$ has a Fatou property if there exists a constant $C>0$ such that the following is true:
If $\{f_{j}\}_{j\geq 0}$ is any bounded sequence in $E,$ with limit $f\in \mathcal{S'}(\mathbb R),$ then $f$ belongs to $E$ and 
$$\|f\|_{E}\leq C \sup_{j\geq 0} \|f_{j}\|_{E}.$$
My Question is:

Is Fatou property true for general normed space(Or Banach Space) ? If not,which kind of normed space(Or Banch space) have Fatou property and which normed space (Banach) fails for Fatou property ?

 A: EDIT: A lot of what I said in my original post is wrong, as it turns out :(
The $L^1$ example below shows that your formulation of the Fatou property is not as useful as it first appears.
The Fatou property as you formulate it is not true in general. As a counterexample, consider 
$$
C_0 := \{f : \Bbb{R} \to \Bbb{C} \mid f(x) \to 0 \text{ as } |x|\to\infty \}.
$$
This is a Banach space (if equipped with the sup norm), continuously embedded in $\mathcal{S}'$, but if you take something like
$$
f_n = \sum_{k=1}^n \chi_{[k - 1/k^2, k+1/k^2]},
$$
then (you have to replace the "squares" by "triangles" of course, to make everything continuous) $f_n \to f$ in $L^1$ (and hence in $\mathcal{S}'$) for some $f$ which no longer belongs to $C_0$ (check this).
One natural sufficient condition for what you want is the following. Assume that $E$ can be written as
$$
E = \{f \in \mathcal{S}' \mid \sup_{g\in F} |\langle f,g \rangle | < \infty \}
$$
with the corresponding norm, where $F$ is a suitable subset of $\mathcal{S}'$.
The problem is now that even the space $L^1$ does not satisfy your formulation of the Fatou property. To see this, take any function $f \in C_c$, $f \geq 0$ with $\int f \,dx =1$ and let
$$
f_n(x) := n \cdot f(nx).
$$
It is relatively easy to see that $f_n \to \delta_0$ (delta distribution at the origin) in $\mathcal{S}'$ and that $\Vert f_n \Vert = 1$ for all $n$. But of course $\delta_0 \notin L^1$.
The problem here seems to be that $L^1$ is not weak-$\ast$-closed in $\mathcal{S}'$.
An alternative formulation of the Fatou property (which holds in many interesting cases, especially for $L^1$) can be formulated in the context of so called Banach function spaces. These are spaces which continously embed into $L_\rm{loc}^1$. The Fatou property then states that if $f_n \to f$ pointwise a.e. and if $\Vert f_n \Vert_E$ is bounded, then $f \in E$ with
$$
\Vert f \Vert_E \leq \liminf_n \Vert f_n \Vert.
$$
Of course, this property fails as soon as smoothness-properties (like differentiability) enter.
BTW: Very interesting question.
