(06/04/2021, 01:00 PM)MphLee Wrote: There is a link indeed.
Every Hilbert space \( H \) is a vector space equipped with a special bilinear form, i.e. a homomorphism antilinear in the first variable and linear in the second one, of vector spaces valued in the base field
\( \langle -\,,\,-\rangle_H:H\times H\to \mathbb C \),called inner product, whose induced metric space structure is complete.
Every(modulo some abstract nonsense) Category \( \mathcal C \) comes equipped with a special bifunctor, i.e. a homomorphism of categories, contravariant in the first variable and covariant in the second, valued in the cat. of sets
\( {\rm Hom}_{\mathcal C}(-\,,\,-):{\mathcal C}^{op}\times {\mathcal C}\to {\bf Set} \),called Hom functor, a category can be asked to be complete under some limits of diagrams (eg. every sequence has a categorical limit).
Given two Hilbert spaces \( H_1, H_2 \) and two operators \( H_1\array{\overset{F}{\rightarrow}\\ \underset{G}{\leftarrow}}H_2 \): we say that \( G \) is adjoint of \( F \) if we have the identity
\( \langle f,G(g)\rangle_{H_1}=\langle F(f),g\rangle_{H_2} \)
Given two categories \( \mathcal C, \mathcal D \) and two functors \( {\mathcal C}\array{\overset{F}{\rightarrow}\\ \underset{G}{\leftarrow}}{\mathcal D} \): we say that \( G \) is LEFT adjoint of \( F \) if we have the natural isomorphism
\( {\rm Hom}_{\mathcal C}( X,G(Y))\simeq {\rm Hom}_{\mathcal D}( F(X),Y) \)
So adjunctions in category theory subsume and immensely extend the "classical" setting in functional analysis.
Adjunctions arise everywhere and are probably the most profound, deep and meaningful concept that mathematicians has ever defined.
Some example of adjunctions are:
- betwen algebra and geometry;
- between existential and universal quantification (logic),
- between Necessity and Possibility (modal logic)
- between set of point and discrete topology functor (topology),
- an honorable example of adjunction is the following law in arithmetic and real algebra
\( w^{(\lambda z)}=(w^\lambda)^z \)
- The curryng isomorphism
\( X^{T\times Y}\simeq(X^T)^Y \)I leave you not with an example but with two suggestive views
\( |x-c|_X<\delta(\epsilon)\Rightarrow|f(x)-\lim_c f|_Y<\epsilon \)\( \sigma\bullet S(x)=F(\sigma)\bullet x \)
Yes! That was it; I'm glad I don't have to go searching for it. I'm gonna spend some time fiddling with,
\(
\vartheta(w) = \sum_{n=0}^\infty f(n+1)\frac{w^n}{n!}\\
H\vartheta(w) = \sum_{n=0}^\infty f^{\circ n+1}(1) \frac{w^n}{n!}\\
\)
Where the chain is,
\(
\frac{d^{z-1}}{dw^{z-1}}|_{w=0} \vartheta = f\\
\frac{d^{z-1}}{dw^{z-1}}|_{w=0} H \vartheta = \uparrow f\\
\)
Now \( H \) is not linear; but there may be a way to pull out an adjoint on the inner product,
\(
(f,g) = \int_0^\infty f(x)\overline{g(x)}\,dx/x\\
\)
Where the law is,
\(
(f,x^{\overline{z}}) = \Gamma(z) \frac{d^{-z}}{dw^{-z}}\vartheta\\
(Hf,x^{\overline{z}}) = \Gamma(z) \frac{d^{-z}}{dw^{-z}}H\vartheta\\
\)
And if we had something like \( H^* \) which satisfies,
\(
(Hf,g) = (f,H^*g)\\
\)
Then if we could iterate \( H^* \) on \( x^{\overline{z}} \) we'd have the fractional hyper-operator nonsense.
BTW, I use the convention that the inner product is linear in the first, and antilinear in the second variable. I'm a mathematician, stop this quantum physics nonsense notation.
We are assuming that \( |f(z)| < M \) is bounded for \( \Re(z) > 0 \). We are assuming it takes \( \mathbb{R}^+ \to \mathbb{R}^+ \); and additionally that \( f : \mathbb{C}_{\Re(z) > 0} \to \mathbb{C}_{\Re(z) > 0} \). This is enough for these transforms to converge.