There is a quotation below:
Let $A$ be a Banach space, $\mathbb{B}(A)$ be the space of all bounded linear maps from $A$ to $A$ and $C\subset \mathbb{B}(A)$ be any convex set. If a net $\{T_{i}\}_{i\in I}\subset C$, and the element (k-fold direct sum) $$T(a_{1})\oplus ...\oplus T(a_{k})$$ belongs to the weak closure (take $l^{p}$-norm on $A\oplus...\oplus A$) of $$\{T_{i}(a_{1})\oplus...\oplus T_{i}(a_{k})\}_{i\in I}$$ hence, by the Hahn-Banach Theorem, also to the norm closure of the convex hull.
I do not know how to utilize Hahn-Banach theorem here. Could someone explain to me ?
The relevant consequence of the Hahn-Banach theorem(s) here is that in a normed space $X$, for every convex subset $M$, the weak closure and the norm-closure coincide. Since the norm topology is finer than the weak topology, we always have $\operatorname{cl}_{\lVert\cdot\rVert}(M) \subset \operatorname{cl}_w(M)$, and the Hahn-Banach theorem tells us that we can separate every $x \notin \operatorname{cl}_{\lVert\cdot\rVert}(M)$ from $\operatorname{cl}_{\lVert\cdot\rVert}(M)$ by a continuous linear functional, hence also $x\notin \operatorname{cl}_w(M)$, which means $\operatorname{cl}_w(M) \subset \operatorname{cl}_{\lVert\cdot\rVert}(M)$ and thus the equality.
In the situation here, we have a point $p$ in the weak closure of some set $S$, and hence also in the weak closure of the convex hull $M$ of $S$. Then by the above, it follows that $p$ belongs to the norm-closure of $M$.