I am trying to understand more about the Gelfand-Pettis integral. From wikipedia:

[![enter image description here][1]][1]

What does it mean that $V$ "admits a dual space?" When $V$ is a Banach space, $V^{\ast}$ is taken to be the space of bounded linear functionals on $V$. When $V$ is a smooth (locally constant) representation of a totally disconnected group, $V^{\ast}$ is usually taken to mean the space of smooth linear functionals on $V$.

Is there a general notion of a topological vector space admitting a dual space? Or is this just context dependent? [1]: https://i.sstatic.net/ji7cP.png

I am trying to understand more about the Gelfand-Pettis integral. From wikipedia:

What does it mean that $V$ "admits a dual space?" When $V$ is a Banach space, $V^{\ast}$ is taken to be the space of bounded linear functionals on $V$. When $V$ is a smooth (locally constant) representation of a totally disconnected group, $V^{\ast}$ is usually taken to mean the space of smooth linear functionals on $V$.

Is there a general notion of a topological vector space admitting a dual space? Or is this just context dependent?