So I’m trying to get this all straightened out in my head.
In Quantum mechanics we use a Hilbert Space $\mathcal{H}$ as our vector space and we say that its elements is the set of Ket’s $\left|\psi\right>$ that are our state vectors. The Bra’s $\left<\psi\right|$ are the adjoint’s of $\left|\psi\right>$. The inner product is then $\left<\psi|\psi\right>: \mathcal{H}\times\mathcal{H}\to\mathbb{C}$. I’m all fairly comfortable with this, but please point out some errors because I’m almost positive I made some.
Now my main question. What is the dual of our Hilbert space? Is it just the bra’s? When I look up the definition of a dual space it’s says it’s the set of all linear functionals that take elements from our vector space and map it to the underlying field. So would it be accurate to say that the bra’s are not the dual space but instead the dual space is the set of all functionals $\left<\psi|\cdot \right>$ that takes elements of $\mathcal{H}$ as its argument and spits out elements of $\mathbb{C}$ because it’s an inner product?
Sorry if this is worded weird as I’m not used to rigorous mathematical definitions. Please point out my misunderstandings! Thanks!