Weak operator topologies on $B(X')$

Question

Let $X$ be a Banach space with dual $X'$. On the space of bounded operators $B(X')$ on $X'$ we can define the following two weak operator topologies defined by the seminorms:

1. $T \mapsto |\langle Tx', x'' \rangle| = |x''(Tx')|$ for $x' \in X$, $x'' \in X''$ defined by the dual pair $\langle X', X'' \rangle$ and
2. $T \mapsto |\langle Tx', x \rangle| = |x(Tx')|$ for $x' \in X$, $x \in X$ defined by the dual pair $\langle X', X \rangle$

I think, the first topology is usually referred to as WOT on $B(X')$. Is there also a name for the second topology? Can this topology be identified with some other natural topology on $B(X')$? If not, where can I find some information about this topology?

It also seems that this construction can be generalized to an arbitrary dual pair $\langle X, Y \rangle$ of vector spaces $X$ and $Y$ with some duality $\langle \cdot, \cdot \rangle : X \times Y \to \mathbb{R}$. Then $T \mapsto |\langle Tx, y \rangle|$ is a seminorm for each $x \in X$ and $y \in Y$ and gives a (natural) locally convex topology on the vector space $L(X,Y)$ of all linear mappings $X \to Y$.

Answer 1

This is the so-called weak${}^*$ operator topology (sometimes abbreviated to W*OT). Please see this paper:

Changsun Choi, Ju Myung Kim, Locally convex vector topologies on $B(X, Y)$, J. Korean Math. Soc. 45 (2008), no. 6, 1677-1703.

for more properties of this topology.