What are weakest conditions on a vector space that allow defining dual (adjoint) operator?

I have an infinite dimensional topological vector space $V$ and its dual $V^*$ endowed with a sesquilinear form $$\phi:V^*\times V\to \mathbb{C}: (u,v)\to \phi(u,v) , \qquad u\in V^* , v\in V$$

I've also seen that these structures suffice to construct quantum mechanics. However, at first step I want to define adjoint (conjugate, or dual) operator in this space and get some difficulties.

I've seen the definitions of adjoint operator in Hilbert space, Banach space, and locally convex vector space with strong dual (that imposes weaker conditions on the space).

So I have following questions.

1. What are weakest conditions on the vector space that allows defining dual operators?
2. Are such conditions satisfied in the space $V$ above?
3. Does the strong dual vector space mean that the sequence should converge in strong topology?

Any comment or suggesting reference would be appreciated.

• If $V$ has finite dimension it should work, but as I don't see any topology playing in there, you might be considering infinite dimension? – Dirk May 3 '17 at 8:22
• You're right. The question is mainly about infinite-dimensional vector space. Thanks for your hint. I edited the question. – HR-Physics May 3 '17 at 8:29