How to prove continuity with respect to Mackey topology on dual pairing? In my attempt to properly learn about topological vector spaces, I tried to generalize a well-known concept from Banach spaces to dual pairings:
Let $(X,Y,b)$ be a dual pairing of vector spaces (i.e. $b \colon X \times Y \to \mathbb{R}$ is a bilinear form with respect to which $X$ and $Y$ mutually separate points). Furthermore, let $R\colon X \to Y$ be a positive-definite symmetric mapping, in the sense that $b(x,Ry)=b(y,Rx)$ and $b(x,Rx) \geq 0$ for all $x,y \in X$, with $b(x,Rx) = 0$ if and only if $x=0$. It follows that $R$ is linear and, if I'm not mistaken, continous from $X$ to $Y$ when both are equipped by any polar topology between the weak and Mackey topologies (the same type of topology on both spaces).
Now complete $X$ with respect to the inner product $(x,y) = b(x,Ry)$ to obtain a Hilbert space $H$, so that in particular $X$ is densely contained in $H$.
My question is: Does it follow without any further assumptions on $X$ and $Y$ that the natural embedding $i \colon X \to H$ is continuous, where $X$ is equipped with the Mackey topology $\tau(X,Y)$?
Using heavy machinery, I think this is true when $(X,\tau(X,Y))$ is metrizable as a well-known consequence of Banach-Steinhaus (on separate continuity vs joint continuity of bilinear forms). Does the conclusion hold in full generality?
Thanks a lot in advance!
 A: The answer to the question about continuity of the embedding is negative in its most general form, and in fact  it is not hard to  characterize precisely when
this is so.
But before that,  it is interesting to rephrase the hypotheses in a different way.

*

*As indicated in the question, $X$ is a pre-Hilbert space, so there is no harm in stating this as hypothesis number 1.

Next,  notice that each $y$ in $Y$ defines a linear functional
$$
  \varphi _y: x\in  X\mapsto b(x,y)\in {\bf R}.
  $$
In addition,  the correspondence
$y\mapsto \varphi _y$ is injective, thanks to the fact that the duality $b$ mutually separates points.


*So, instead of emphasizing the bilinear form $b$, we make the hypothesis that we are given a linear subspace $Y$ of the algebraic dual $X^\dagger$.

For each $y$ in $X$, the linear functional $R(y)$ is given by
$$
  R(y)|_x = \langle x,y\rangle , \quad\forall x\in  X,
  $$
so that $R(x)$ is precisely the canonical image of $x$ within $X^\dagger$ given by the inner-product.


*We may therefore add to our new set of hypothesis that  the given subspace $Y\subseteq X^\dagger$ contains the canonical
image of $X$ in $X^\dagger$.

En passant, the above picks up the part of the original set of hypothesis where $b$ is required to separate points in
its first variable.
There is nothing to guarantee that the functionals in $Y$ are continuous on $X$
(relative to its pre-Hilbert space structure), so we shall denote by $Y_c$ the subset of $Y$ formed by the continuous
linear functionals.  Notice that
$$
  X\subseteq Y_c = X^*\cap Y \subseteq Y,
  $$
where the $X$ in the LHS is to be interpreted as the canonical image of $X$ in $X^\dagger$, and $X^*$ refers to the
topological dual (which happens to be naturally isomorphic to $H^*$).
We are therefore ready to state the main result:
Theorem The inclusion $\iota :X\to H$ is continuous with the Mackey topology on $X$, and the norm topology on $H$, iff
$X^*\subseteq Y$ or,  equivalently,  $Y_c=X^*$.
Proof.  Assuming continuity, pick any $\varphi $ in $X^*$,  and let $\tilde \varphi $ be the unique continuous linear functional on
$H$ extending $\varphi $.
The composition
$\require{AMScd}$ \begin{CD}  X @>{\iota}>> H@>{\tilde\phi}>> \mathbb R\\  \end{CD}
which is nothing but $\varphi $,
is therefore continuous wrt the Mackey topology $\tau (X, Y)$.  It is well known that the dual of $X$ relative to $\tau (X, Y)$
coincides with $Y$ (actually $\tau (X, Y)$ is the finest locally convex topology on $X$ whose dual is $Y$),  so we conclude that $\varphi $ lies in $Y$.
Conversely, assuming that $X^*\subseteq Y$, we immediately see that $\iota $ is weakly continuos (i.e. continuos relative to
$\sigma (X,Y)$ and $\sigma (H,H^*)$), and it is not hard to see that this implies continuity relative to
$\tau (X,Y)$ and $\tau (H,H^*)$, the latter topology being precisely the norm topology on $H$.  QED
In order to present a concrete counter-example,
let $X$ be the space of all sequences $x=(x_n)_{n\in {\bf N}}$ of real numbers with finitely many nonzero coordinates, and let
$Y=X$.  Define a duality between $X$ and $Y$ by
$$
  b(x,y)=\sum_{n\in {\bf N}}x_ny_n,
  $$
and let $R$ be the identity map, so that the inner-product
$
  \langle x,y\rangle =b(x,Ry)
  $
is the usual one,  and hence $H=\ell ^2$.
With the conventions above, we have that $Y$ is a proper subspace of $X^*$, so  $\iota $ is not continuos relative to $\tau
(X,Y)$ and the norm topology on $H$.
