Reproducing kernel Hilbert spaces and the isomorphism theorem

A duplicating kernel Hilbert room is a Hilbert room in which the analysis useful

$L_x : f \rightarrow f(x)$ is continual. By connection, the Riesz depiction theory claims that this useful can be stood for as an internal item.

I sense there is something basic I've misconstrued below. If any kind of 2 actual Hilbert spaces of the very same cardinality are isomorphic, after that why is it that $l_2$ is a RKHS, yet $L^2[0,1]$ is not?

2022-06-07 14:31:01
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To complete what Theo and also Jonas currently claimed: Two (facility or actual) Hilbert spaces are isomorphic if they have orthonormal bases with the very same cardinality, as Hilbert spaces. So, every declaration that takes advantage of the Hilbert room framework just, and also holds true or incorrect for one room, will certainly hold true or incorrect for the various other.

Yet a concrete Hilbert room might have extra framework than simply the Hilbert room framework. When you consider the declaration "A duplicating bit Hilbert room is a Hilbert room in which the analysis useful " after that the "evaluation functional" - component infers that the Hilbert room present has (actual or intricate valued) operates as components. This is an added building that some Hilbert spaces have and also some have not.

The room $L^2[0, 1]$ as an example contains equivalence courses as opposed to features, and also the "evaluation functional" can not be well specified due to the fact that it relies on the rep of an equivalence class $[f]$. Actually, for any kind of $x \in \mathbb{R}$ and also for every single actual number $y$ consisting of $\infty$ and also $-\infty$, every equivalence class has a component $f$ such that $f(x) = y$. And also I can additionally specify an abstract facility Hilbert room by claiming that it is the room extended by an orthonormal basis $(e_n)_{n \in \mathbb{N}}$. Currently one can not understand the term "evaluation functional", due to the fact that the components of this Hilbert room are not features.

On the various other hand, the Hardy room of the device disk (Wikipedia) contains holomorphic features, consequently the analysis useful is well specified. It is feasible to confirm that it is additionally continual, yet the evidence takes advantage of the reality that the components of this Hilbert room are holomorphic features on the device disk, which, as I claimed in the past, is added framework that takes place to exist for the Hardy room.

Every one of these instances are isomorphic as separable facility Hilbert spaces, yet this isomorphism does not claim anything concerning any kind of framework that might exist past the Hilbert room framework.

2022-06-07 14:59:44