Lower Semicontinuity Concepts Let $X$ be a real Banach space, let $f:X\rightarrow \overline{\mathbb{R}}$ be a functional. We have known that:


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*If $f$ is weakly lower semicontinuous then $f$ is weakly sequentially lower semicontinuous;

*If $f$ is weakly sequentially lower semicontinuous then $f$ is lower semicontinuous.
a) I would like to construct two counterexamples to show that the reverse is not true.    
(It is interesting to find counterexamples in Hilbert space.)
b) When do these concepts coincide?
Thank you for all help and comments.
 A: b) My Google responses on "weakly sequentially lower semicontinuous" query yield something about convexity. For instance, if $X$ is a normed space, and $f:X\to\overline{\mathbb R}$ is quasi-convex and (strongly) lower semicontinuous then $f$ is also weakly sequentially lower semicontinuous [1, Ex.3]. You can try to google for more such facts. 
[1] Peter Holthe Hansen, 01716 Advanced Topics in Applied Functional Analysis,
Assignment 10.
a) It seems the following. 


*

*An example of a continuous  (and, hence, a lower semicontinuous) function $f:\ell_2\to\mathbb R$ such that $f$ is not weakly sequentially lower semicontinuous. For each point $x_0\in\ell_2$ define a function $f_{x_0}:\ell_2\to\mathbb R$ by putting $f_{x_0}(x)=\min\{\|x-x_0\|-1/3,0\}$ for each $x\in\ell_2$. For each natural number $n$ let $e_n$ be the standard $n$-th ort of the space $\ell_2$. Put $f=\sum_{n=1}^\infty f_{e_n}$.
Then $f$ is continuous as a sum of a family of continuous functions with the locally finite  family of the supports. From the other side, $f$ is not weakly sequentially lower semicontinuous, because $f(0)=0$, but the sequence $\{e_n\}$ weakly converges to $0$ and $f(e_n)=-1/3$ for each $n$.

*An example of a weakly sequentially lower semicontinuous function $f:\ell_2\to\mathbb R$ such that $f$ is not weakly lower semicontinuous.

Maybe we should first to construct a subset $A$  of the space $\ell_2$ such that $0$ belongs to the weak closure of $A$, but $0$ does not belong to the weak sequential closure of $A$. 

I found such a set  $A=\{x_{nm}\}$ in  Bill Johnson’s answer to Question “A point in the weak closure but not in the weak sequential closure”. The zero is the unique not weakly isolated point of the set $A$. So we may define a function $f:\ell_2\to\mathbb R$ by putting $f(0)=-1$, $f|A\setminus\{0\}=-2$ and $f|\ell_2\setminus A=0$.
