Takesaki theorem 2.6 I originally posted this question on MSE and didn't get a satisfactory answer, even after putting a bounty on it. Hence, I thought I should ask here:
Consider the following theorem in Takesaki's book "Theory of operator algebras". In particular, I'm focussed on understanding (ii). It is not mentioned in the picture, but $S$ is the unit ball of $\mathscr{L}(\mathfrak{h})$.

I can see that (ii.1) $\iff$ (ii.2) $\iff$ (ii.3) $\iff$ (ii.4) and that (ii.5) $\implies$ (ii.6) $\implies$ (ii.7). Also, since the ‘weaker’ topologies agree with their $\sigma$-counterparts on bounded subsets, it is also clear that (ii.$x$) $\implies$ (ii.$x+4$) for $x=1,2,3$.
However, why do the other implications hold? The author just says that this follows by general theory of topological vector spaces, so if anyone can fill in the gap that would be great. In particular, I think it will be sufficient to know how I can prove (ii.7) $\implies$ (ii.3).
 A: If one reads Takesaki's proof carefully, we find:

and there (to my reading) is no claim that part (ii) uses "general duality theory".  So let's look at Lemmas 2.4 and 2.5.  2.4 says that $\sigma$-strong$^\ast$ continuous functionals are $\sigma$-weakly continuous.  2.5 says (in particular) that on $S$ the $\sigma$-strong$^\ast$ and strong$^\ast$ topologies agree.  So I guess the logic of the proof of (ii.7)$\implies$(ii.3) is that:

*

*$\omega$ is strong$^\ast$ continuous on $M\cap S$

*So by Lemma 2.4, $\omega$ is $\sigma$-strong$^\ast$ continuous on $M\cap S$

*Somehow now argue that $\omega$ is $\sigma$-strong$^\ast$ continuous on all of $M$, which is exactly (ii.3)

I do not see how to do the last step.

However, on reading around, I found a very similar argument in Stratila, Zsido, Lectures on von Neumann algebras (link to 2nd edition; I am reading the 1st edition).  Indeed, Lemma 1.2 is similar (but not directly applicable).  Let me adapt the argument.
Recall that the seminorms defining the $\sigma$-strong$^\ast$ topology are (mixing my notation with Takesaki's)
$$ x\mapsto \|x\|_u := \big( p_u(x)^2 + p_u(x^*)^2 \big)^{1/2} $$
where $u\in\mathcal L_*(H)$ is positive.  In the arguments which follow, for a general locally convex space, we would need usually need to take sums of seminorms.  However, given $u_1,\cdots,u_n$ positive, there is $u$ positive with $\sum_{k=1}^n \|x\|_{u_k} \leq \|x\|_u$ for $x\in M$, and so taking sums is not necessary, in this special case.
We have $\omega$ which is $\sigma$-strong$^\ast$ continuous on $M\cap S$.  Thus $\omega$ is norm continuous on $M\cap S$ and hence norm continuous (so bounded) on $M$.  Given $\epsilon>0$, the set $\{ x\in M\cap S : |\omega(x)|<\epsilon \}$ is $\sigma$-strong$^\ast$ open (in the relative topology on $M\cap S$) and so there is $u\in\mathcal L_*(H)$ positive with
$$ x\in M, \|x\|\leq 1, \|x\|_u < 1 \implies |\omega(x)|<\epsilon. $$
For any non-zero $y\in M$, let $x = k^{-1} y$ where $k > \max(\|y\|, \|y\|_u)$, so that $\|x\|\leq 1$ and $\|x\|_u<1$.  Thus $|\omega(x)|<\epsilon$, that is, $|\omega(y)|<\epsilon k$.  We can choose $k$ so that $k \leq \|y\| + \|y\|_u$, so that
$$ |\omega(y)| \leq \epsilon \|y\| + \epsilon \|y\|_u. $$
We now directly use Lemma 1.1 from Stratila and Zsido to find linear functionals $\omega_1,\omega_2$ on $M$ with $\omega = \omega_1+\omega_2$ and
$$ |\omega_1(y)| \leq \epsilon \|y\|, \quad |\omega_2(y)| \leq \epsilon \|y\|_u \qquad (y\in M). $$
Hence $\|\omega - \omega_2\| = \|\omega_1\| < \epsilon$ and $\omega_2$ is $\sigma$-strong$^\ast$ continuous.
Conclude that $\omega$ is in the norm closure of the set of $\sigma$-strong$^\ast$ continuous linear functionals.  By Takesaki, Lemma 2.4, this set is simply $\mathcal L_*(H)$ (which is norm closed) and so (ii.7)$\implies$(ii.1) (which implies (ii.3)).

To show S+Z Lemma 1.1 consider $M\oplus M$ and the seminorm $(a,b)\mapsto \epsilon\|a\| + \epsilon\|b\|_u$, and the linear functional defined on the subspace $D = \{ (a,a) : a\in M \}$ by $(a,a)\mapsto \omega(a)$.  Now Hahn-Banach this functional to all of $M\oplus M$.
(In the proof of the relevant part of Lemma 1.2 a factor of $\|\varphi\|$ appears, but I don't see why this isn't $\epsilon$.)
A: This is covered in the the monograph „Saks Spaces and Applications to Functional Analysis“.  The short version is that one can use the fact that all closed subspaces (in particular, f.d. ones) of a Hilbert space  are nicely complemented to express $L(H)$ as a projective  limit of special cases of operators on finite dimensional spaces.  There are three ways to use this to give  von Neumann algebras suitable topologies (using the categories of Banach spaces, Saks spaces and locally convex spaces, respectively) and these have the bounded linear functionals, the ones in your second list and, finally, those in the third one as duals.  This, plus simple results on duality for l.c.s.‘s, gives the result.  The details are in the fourth chapter of the above reference.
Added as edit: the results and methods cited in the above text (which are based on a Comptes Rendues article „Topologies sur l’espace des opérateurs dans l’espace hilbertien“, A 276 (1973) 1509-1511) are more precise than those cited here.  They are based on three families of l.c. topologies on $L(H)$
a) the norm topology;
b) the weak, the strong and the strong-$\ast$ operator topologies $\tau_w,\tau_s,\tau_{s\ast}$;
c)  the weak, strong and strong-$\ast$-strict topologies $\beta_w,\beta_s,\beta_{s\ast}$ which are the finest l.c. topologies which agree with those of b) on the unit ball.
The relations with duality lie in the fact that the corresponding dual spaces are
a) the bounded linear forms—this has no natural representaion as a space of operators;
b)  the space of finite rank operators;
z
c)  the space of nuclear operators.
In the latter two cases, the duality is realised by the trace functional.  These are equivalent to the representations of the duals in Takesaki, but are more natural and transparent in the context of l.c. duality.
The topologies of b) are closely related to the ultraweak and ultrastrong topologies of von Neumann but it can be argued that they have better properties. For example, they are complete.   A major result is that they are natural topologies of the duality between the spaces of bounded, resp. nuclear operators—those of uniform convergence on the compact, resp. weakly compact subsets (i.e. the Mackey topology).
As mentioned above, the proofs are quite transparent and are based on using category theory techniques (duality of projective limits) to reduce to the finite dimensional cases.
A: (TL, DR). Modulo the duality theory of Banach spaces, this simply amounts to null-sequences together with their limit, $0$, are compact, which is really trivial. But I agree that Takesaki probably could've made the exposition more clear by mentioning the Krein-Smulian theorem which is used implicitely so many times here.
Below is the elaboration of the above.
I've struggled with the same question when I first read this part of Takesaki some years ago. While Mathew's answer is great, I think what is more helpful here is a discussion of the Krein-Smulian theorem, which will be used immediately again in part (iv) of that same theorem.
Theorem (Krein-Smulian). Let $X$ be a Banach space, $X^*$ the dual space of continuous linear functionals on $X$, $S^*$ the closed unit ball of $X^*$, $C$ a convex subset of $X^*$, then $C$ is $\sigma(X^*, X)$-closed (aka. weakly-* closed) if and only if for all $r > 0$, the intersection $r S^* \cap C$ is $\sigma(X^*, X)$ closed.
This is a nontrivial theorem, and unfortunately seems not so well-known as it should be in the operator algebra community, especially for the young generation. If I recall correctly, Dixmier's classical treatise on von Neumann algebra also dealt the issue in the same way by referring a more generalized version of the Banach-Smulian theorem in Bourbaki (discussed below). This theorem, in the form stated above, is proved in various places, e.g. Pedersen's Analysis Now, Theorem 2.5.9 on page 74 (the statement is slightly different, but is equivalent to the form stated above by the theorem of Banach-Alaoglu which is much easier). But in many places, the techniques used in the proof of this theorem seems a bit ad-hoc. But even with the "ad-hoc approach" as is done e.g. in Pedersen's book, there's already a glimpse of the duality theory in the form of the polar of $r^{-1}S$ (closed ball of $X$ with radius $r^{-1}$) is $rS^*$, and the bipolar theorem is the pillar of the duality theory (of locally convex spaces).
This being said, allow me to say a few words about how to make the duality theory more apparent in this issue, thus making the seemingly hard proof of Krein-Smulian almost trivial. The treatment of the same theorem in Dunford-Schwartz (V.5.3-V.5.7) is very much in this spirit. Here, one first introduces a new locally convex topology on $X^*$, called $BX$-topology in Dunford-Schwartz, which is in fact exactly the topology $\tau_{ns}$ of convergence on null-sequences, i.e. a neighborhood basis of $0$ in $(X^*, \tau_{ns})$ is given by the polars of the null-sequences in $X$ (this is essentially Lemma V.5.4 in Dunford-Schwartz, which is the key of the proof given there). Note here that all null-sequences are precompact as sets. Now the Krein-Smulian theorem can be restated as on $X^*$, the topology $\tau_{ns}$ and $\sigma(X^*, X)$ (which is coarser than $\tau_f$, the topology of converge on finite sets in $X$ on bounded parts, or simple convergence on bounded parts, this is the $BX$-topology in terms of Dunford-Schwartz) yield the same topological dual of $X^*$. It is well-known in the duality theory of Banach spaces (or more generally, locally convex spaces) that this holds if and only if $\tau_{ns} = \tau_f$ is finer than $\sigma(X^*, X)$, which is obvious in our case, and is coarser than the Mackey topology $\tau_c$ of convergence on compact sets in $X$, which is also easy since all null sequences are precompact (If I am being nit-picking, the Mackey topology should be the topology of convergence on convex compact sets, which is not really a problem here due to another theorem of Krein-Smulian, saying that the closed convex hull of a weakly compact set in a Banach space is still weakly compact, which is not true in general locally convex spaces).
In the above spirit, one can in fact prove a much stronger version of the Krein-Smulian theorem. Note that the topology $\tau_f$, $\tau_{ns}$, $\tau_c$ on a topological dual $E^*$ of a locally convex space $E$ still make sense (with a slight caveat that $\tau_f$ might be better described using equi-continuous sets in this general case).
Theorem (Banach-Dieudonné-Krein-Smulian) If $E$ is metrisable, then $\tau_f = \tau_{ns} = \tau_c$ on $E^*$.
Note that we merely need the case where $E$ is Banach and $\sigma(X^*, X) \subseteq \tau_f = \tau_{ns} \subseteq \tau_c$ to prove Krein-Smulian (modulo the well-known duality theory involving the Arens-Mackey topology as discussed above).
Interestingly, this Banach-Dieudonné-Krein-Smulian is actually quite easy to state and prove, in the sense that it is very natural to consider along the lines of a systematic duality theory of locally convex spaces. The earliest systematic reference for this is of course Bourbaki's Espaces vectoriels topologiques as referenced by Diximier in his book on von Neumann algebras. I understand well that people might frown upon Bourbaki because it is no light reading. I recently find the book A course on topological vector spaces by Jürgen Voigt explains all this quite well in merely about 100 pages. It also includes a much more natural line of consideration, due to Grothendieck (which is also in Bourbaki but is much heavier to read), for the Eberlian-Smulian theorem, which will be used in a crucial way in section 5, Chapter III of Takesaki when he gives characterization theorem (Theorem 5.4) of $\sigma(M_*, M)$-precompact sets in the predual $M_*$ of a von Neumann algebra $M$, which is crucial in his treatment of traces on von Neumann algebras (Theorem 2.4 in Chapter V), and in various other places, which you might also be interested later.
Finally, IMHO, as one who struggled a lot on reading Takesaki's volumes, I must confess often times, I find Takesaki's treatment is not the most economic nor most elucidating, e.g. the book of Stratila-Zsido mentioned by Mathew often treats the same subjects better from a pedagogical point of view. This being said, I find that Takesaki's volumes are often the only place where I can consistently find a satisfactory answer to many basic natural questions in the basics of theory of operator algebras, often in hindsight after I've read similar, more pedagogically friendly treatment elsewhere. I say from my experience that Takesaki's volumes are probably not the ideal place for most to learn operator algebras for the first time, but if one is familiar with these books, one certainly will appreciate them more and more as time goes by. As for a practical prerequisites on functional analysis to read volume 1 of Takesaki, logically speaking, Chapter V of Dunford-Schwartz is probably sufficient + the fixed point theorem of Ryll-Nardzewski, which is excellently covered in the appendix of Stratilo-Zsido, who followed the 1967 article of Namioka & Asplund. But to "naturalize" the deep theorems in these volumes probably demands a deeper dive into the duality theory of locally convex spaces, among many other things.
