Nets and closedness of $\mathcal{B}(H)$ in the topology of pointwise convergence

Let $$H$$ be an infinite-dimensional Hilbert space and $$\mathcal{B}(H)$$ the set of bounded linear operators on $$H$$.

One way to define the strong operator topology (SOT) on $$\mathcal{B}(H)$$ is by specifying how nets converge: if $$\{T_\lambda\}_{\lambda\in\Lambda}$$ is a net in $$\mathcal{B}(H)$$ and $$T\in\mathcal{B}(H)$$, then $$T_\lambda\to T$$ if $$T_\lambda(x)\to T(x)\tag{1}\label{convergence}$$ in $$H$$ for all $$x$$.

This appears to be related to the general notion of "topology of pointwise convergence" for a family $$F$$ of functions $$X\to Y$$, where $$X$$ and $$Y$$ are topological spaces. This is the subspace topology that $$F$$ inherits from the space of all functions $$Y^{|X|}$$ equipped with the product topology.

The following two questions are kind of sanity checks to see if I've understood things correctly.

Question 1: Is the SOT on $$\mathcal{B}(H)$$ the topology of pointwise convergence on the set $$\mathcal{B}(H)\subseteq H^{|H|}$$, where $$H$$ is equipped with the norm topology?

Now let $$\mathcal{L}(H)$$ be the set of all linear maps $$H\to H$$ (not necessarily bounded). Then we could also put a "strong operator topology" on $$\mathcal{L}(H)$$ in the same way, using $$\eqref{convergence}$$. To distinguish this topology, let's call it SOT$$_{\mathcal{L}(H)}$$. Then it seems that the SOT on $$\mathcal{B}(H)$$ is really the subspace topology inherited from $$(\mathcal{L}(H),\text{SOT}_{\mathcal{L}(H)})$$.

Question 2: Is $$\mathcal{B}(H)$$ closed in $$(\mathcal{L}(H),\text{SOT}_{\mathcal{L}(H)})$$?

I suspect the answer is no, from looking at this answer.

Remark: I wanted to check this because it seems to be a slightly subtle fact that (by the uniform boundedness principle), pointwise convergence (at all $$x\in H$$) of a sequence $$T_n\in\mathcal{B}(H)$$ implies that the operator $$T$$ defined by $$T(x)=\lim_{n\to\infty}T_n(x)$$ is bounded, whereas this is not true for a general net $$\{T_\lambda\}_{\lambda\in\Lambda}$$. So one cannot really detect boundedness of the operator $$T$$ using nets as in \eqref{convergence}, but rather it should be thought of as a characterisation of convergence of nets when one already knows the limit $$T$$ is bounded.

Answer to question 1:

Yes. A net $$f_\alpha$$ is convergent pointwise to some $$f$$ iff $$f_\alpha(x)\to f(x)$$ for all $$x$$. This is precisely how you have characterised SOT.

Answer to question 2:

No. Let $$L$$ be an arbitrary linear map. Let $$\mathcal V$$ denote the set of finite dimensional subspaces of $$H$$. The relation $$V\subseteq W$$ makes $$\mathcal V$$ into a directed set as can be checked easily. For any $$V\in\mathcal V$$ let $$p_V$$ denote the orthogonal projection to $$V$$.

Then the net $$L_V:= p_VLp_V$$ converges pointwise to $$L$$, since for any $$x\in H$$ there is a $$V\in\mathcal V$$ with $$x\in V$$ and $$L(x)\in V$$, then for all $$W$$ with $$V\subseteq W$$ you have $$L_W(x)=L(x)$$ and $$L_V\to L$$ pointwise.

Further if you write $$H=V\oplus V^\perp$$ then $$L_V=p_VL p_V$$ operates only on the first factor, which is finite dimensional. So $$L_V$$ is continuous.

So $$\mathcal B(H)$$ is dense in $$\mathcal L(H)$$ in pointwise convergence (infact this shows already that the finite rank linear maps are dense).