Intereting Posts

Number of real roots of the equation $2^x = 1+x^2$
Find the value of : $\lim_{x \to \infty}( \sqrt{4x^2+5x} – \sqrt{4x^2+x})$
Tarski-like axiomatization of spherical or elliptic geometry
If U1, U2, U3 are iid Uniform, then what is the probability of U1+U2>U3?
Prove that $\log _5 7 < \sqrt 2.$
Does this really converge to 1/e? (Massaging a sum)
Neighborhood base at zero in a topological vector space
Intuition for the Universal Chord Theorem
Is it misleading to think of rank-2 tensors as matrices?
Is the product of three positive semidefinite matrices positive semidefinite
(ZF) Prove 'the set of all subsequential limits of a sequence in a metric space is closed.
Does an infinite random sequence contain all finite sequences?
What is the $x$ in $\log_b x$ called?
How does (21) factor into prime ideals in the ring $\mathbb{Z}$?
How to visualize a rank-2 tensor?

Let $(X,s)$ be a topological vector space over $\mathbb{F}$ with linear topology $s$, which we will henceforth refer to as the *strong topology*. Then, as usual we can construct the continuous dual space $(X,s)’$, which consists of all linear maps $X \rightarrow \mathbb{F}$ which are continuous with respect to the strong topology. Now, we can construct the weak topology $w$, the smallest topology on $X$ such that every map $\Lambda \in (X,s)’$ remains continuous.

So now here’s my question. The weak topology on $X$ yields a new topological vector space $(X,w)$, so we can construct a continuous dual for that too. So what does $(X,w)’$ look like? It must at the very least contain $(X,s)’$, of course. Other possible questions:

- For what types of TVS’s can we infer something about the structure of $(X,w)’$ with respect to $(X,s)’$? What happens in a normed vector space?
- Can somebody give me an explicit construction indicating that there exists a linear map $X \rightarrow \mathbb{F}$ which is not continuous in the strong topology but is continuous in the weak topology?
- I suppose we can repeat this process to produce a “doubly weak” topology $ww$ on $X$ that is the weakest topology such that every $\Lambda \in (X,w)’$ remains continuous. Then we can produce yet another continuous dual space $(X,ww)’$. Does this process repeat forever? Does it terminate?

- Derivative of convolution
- Finite dimensional subspace of $C()$
- Which functions are tempered distributions?
- Differentiation continuous iff domain is finite dimensional
- Isomorphic Hilbert spaces
- How common is it for a densely-defined linear functional to be closed?

- On continuously uniquely geodesic space
- How to prove that the implicit function theorem implies the inverse function theorem?
- Taking the topological dual in terms of category theory
- On the weak and strong convergence of an iterative sequence
- Proof of equicontinuous and pointwise bounded implies compact
- Why $C_0^\infty$ is dense in $L^p$?
- Operator: not closable!
- The norm and the spectrum of $B(L^p(X,\mu))$
- $d(w, V) = 1$ and $U = V \oplus\{\lambda w :\lambda \in \mathbb{F}\} $.
- Fredholm Alternative as seen in PDEs, part 2

If $F$ is a family of linear functionals on a vector space $X$ then it induces a weak topology $w_F$ on $X$. The dual space of $(X,w_F)$ is the linear span of $F$ in the algebraic dual of $X$. In particular $(X,s)’ = (X,w)’$ always holds.

This fully answers the first bullet. It also shows that there is no example as in the second bullet and that the procedure in the third bullet stabilizes at $(X,w)$ after one step.

The point is that if $\varphi \colon X \to \mathbb{F}$ is $w_F$-continuous then $U = \{x \in X \mid \lvert \varphi(x)\rvert \lt 1\}$ is a $w_F$-open neighborhood of $0$. By definition of the weak topology, there are $f_1,\dots,f_n \in F$ and $\varepsilon \gt 0$ such that $V = \bigcap_{i=1}^n \left\{x \in X \mid \lvert f_i(x)\rvert \lt \varepsilon\right\} \subseteq U$ and in particular $\bigcap_{i=1}^n \ker f_i \subseteq \ker \varphi $. Therefore $\varphi$ is a linear combination of the $f_i$ (a proof of this last assertion is here).

You can find a detailed proof in pretty much every book on topological vector spaces. A good reference is chapter 3 of Rudin’s *Functional Analysis*, see in particular Lemma 3.9 and Theorem 3.10 (the Hausdorff assumption is only there because Rudin assumes locally convex spaces to be Hausdorff, as many authors do).

- Slightly changing the formal definition of continuity of $f: \mathbb{R} \to \mathbb{R}$?
- Is $f(x,y)=(xy)^{2/3}$ differentiable at $(0,0)$?
- General Formula for Equidistant Locus of Three Points
- Intuition behind Cantor-Bernstein-Schröder
- Presentation of Rubik's Cube group
- Is there an infinite number of primes constructed as in Euclid's proof?
- Prove $ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $
- Necessary condition for have same rank
- {5,15,25,35} is a group under multiplication mod 40
- An short exact sequence of $\mathfrak{g}$ of which head and tail are in category $\mathcal{O}$.
- The $p$-adic integers as a profinite group
- Rational roots of polynomials
- $\Bbb Z_m \times \Bbb Z_n$ isomorphic to $\Bbb Z_{\operatorname{lcm}(m,n)}\times \Bbb Z_{\gcd(m,n)}$
- A combinatorial proof of Euler's Criterion? $(\tfrac{a}{p})\equiv a^{\frac{p-1}{2}} \text{ mod p}$
- how does expectation maximization work?