I have a question regarding dual spaces.
Before, let me write that this all issue looks really problematic to me, and I already touched it quickly in another question. However, in that occasion, the focus was slightly different and I was advised to ask a separate question. Thus, here there is my new question.
Question:
Given a topological vector space $X$, how do we know (decide?) what is its dual?
Do we get this information through the Riesz Representation Theorems?
If we have a couple of dual spaces, are they interchangeable (i.e. one is the dual of the other)?
As always, any feedback or explanation will be greatly welcome.
Thank you for your time.
PS: I do hope the question does not look too naive, but being self-thaught (I know I tend to repeat this too often, but I think it is relevant) I think I have exactly the kind of questions concerning the general intuition and relation between different objects that are usually addressed in a class by professors or tutors in the flesh.
Algebraic Dual
Given a vector space.
Consider its algebraic dual: $$V^*:=\mathcal{L}(V,\mathbb{C})$$ (Remark: This won't be of to much interest!)
Topological Dual
Given a topological vector space.
Consider its topological dual: $$V':=\mathcal{L}(V,\mathbb{C})\cap\mathcal{C}(V,\mathbb{C})$$ (It obtains algebraic structure by pointwise operations.)
Regard the dual pairing: $$V'\times V:\quad(f,v):=f(v)$$
Introduce its evaluations: $$\varepsilon_v(f):=(f,v)=f(v)$$
It can be endowed with the dual topology: $$\mathcal{T}(V'):=\tau\left(\bigcup_{v\in V}\varepsilon_v^{-1}\mathcal{T}(\mathbb{C})\right)$$ (Warning: This won't be the only topology on the dual!)
Bounded Dual
Given a Banach space.
Consider its bounded dual: $$E':=\mathcal{L}(E,\mathbb{C})\cap\mathcal{B}(E,\mathbb{C})$$ (It is identical to the topological dual!)
It is a Banach space itself with: $$\|f\|:=\sup_{x\in E:\|x\|\leq1}|f(x)|$$ (Note: This gives another topology on the dual!)
Hilbert Dual
Given a Hilbert space.
Riesz identifies the dual space by: $$\delta_\eta\varphi:=\langle\eta,\varphi\rangle:\quad\Phi:\mathcal{H}\to\mathcal{H}':\eta\mapsto\delta_\eta$$ (Caution: The scalar product is another dual pairing!)
This identification is antilinear: $$\varepsilon_{\varphi+\psi}=\varepsilon_\varphi+\varepsilon_\psi\quad\varepsilon_{\lambda\varphi}=\overline{\lambda}\varepsilon_\varphi$$
Especially, it is an isometry: $$\|\varepsilon_\varphi\|=\|\varphi\|$$ (Hint: It allows a noncanonical linear isometry!*)
Measure Dual
In a new thread: Riesz-Markov-Kakutani
Reference
*See the thread: Riesz: Isomorphy