Does "two topological vector spaces in duality" means they are each other's continuous dual?

Question

When saying two topological vector spaces $E$ and $F$ are in duality, does it mean that they are each other's continuous dual, i.e. $E = F^*$ and $F=E^*$, or just that one is the other's continuous dual, not necessarily true for the reverse?

If it is the former, when is $E = F^*$ and $F=E^*$ true?

For example,

a biorthogonal system is a pair of topological vector spaces $E$ and $F$ that are in duality, with a pair of indexed subsets $ \tilde v_i$ in $E$ and $\tilde u_i$ in $F$ such that $$ \langle\tilde v_i , \tilde u_j\rangle = \delta_{i,j} $$ with the Kronecker delta

Thanks and regards!

Answer 1

You might find the paragraph starting out with "Duality in the theory of topological vector spaces." helpful here:

Duality. Encyclopedia of Mathematics. URL: https://www.encyclopediaofmath.org/index.php?title=Duality&oldid=21977

Answer 2

Let $E,F$ be two real vector spaces. A "pairing" is a map $E \times F \to \mathbb R$, written say $\langle x,y \rangle$, that is linear separately in each variable, and such that each space separates points in the other: for each nonzero $x \in E$ there is $y \in F$ with $\langle x,y \rangle\ne 0$ and for each nonzero $y \in F$ there is $x \in E$ with $\langle x,y \rangle \ne 0$. We say "$E$ and $F$ are in duality".

Now define the weak topology $\sigma(E,F)$ on $E$ as the weakest topology such that, for each fixed $y \in F$, the map $x \mapsto \langle x,y \rangle$ is continuous. Then we may identify $F$ with the set $E^*$ of all $\sigma(E,F)$-continuous linear functionals on $E$ as follows: continuous linear functional $\phi \in E^*$ corresponds to point $y \in F$ iff $\phi(x) = \langle x,y \rangle$ for all $x$. Similarly, define the weak topology $\sigma(F,E)$ on $F$, and then identify the $\sigma(F,E)$-dual $F^*$ of $F$ with $E$.