I think I may be on the way to answering this question. I've always wondered if Hilbert spaces could be useful for Tetration. This may be the first case I've ever seen it viable. I'll just post what I have here, and the rest of my thoughts I will compose in a PDF. This is an attempt at placing Kneser's tetration in a Hardy space, and deriving uniqueness/existence in this manner. Avoiding the Riemann mapping theorem only in technicality. I'll be brief here.

Let's call the space of holomorphic functions,

For . Where the sum is infinite or finite; so long as we are guaranteed convergence. We'll define the inner-product as, for ,

This produces a hilbert space where,

Where if we get , if we get a finite real number; and otherwise . To those of you who may have fiddled with Hilbert Spaces, this is isomorphic to the space of almost periodic functions, if we look at it only in terms of rays for , this is precisely the space of almost periodic functions. This is a space, not exactly a familiar kind of Hilbert Space, but a non-separable Hilbert space. Nonetheless, we can now translate from one Hilbert space to the next.

If we call, , where exists in a different space, specifically,

Of which their exists an inner product (This can be explained better with a good understanding of Hilbert spaces; this is just a mapping between spaces). In this new space the inner product looks like,

For an appropriately chosen measure (where again, it requires some work to justify this, but it's something in this vector). This allows us to no longer talk about as a function on its own, but rather talk about other functions which exist in a different Hilbert space, and approximating the solution in the different Hilbert space.

I apologize if I'm being a bit top heavy in my analysis right now. I'm going to start a paper on this, mostly with the goal of finding a different manner of defining uniqueness and expressibility of Kneser's Tetration. Even if I have to use Kneser's tetration to do everything I'm going to do; I think it may be very important to view it as a Hilbert Space, and view the construction in a different Hilbert space.

Let's call the space of holomorphic functions,

For . Where the sum is infinite or finite; so long as we are guaranteed convergence. We'll define the inner-product as, for ,

This produces a hilbert space where,

Where if we get , if we get a finite real number; and otherwise . To those of you who may have fiddled with Hilbert Spaces, this is isomorphic to the space of almost periodic functions, if we look at it only in terms of rays for , this is precisely the space of almost periodic functions. This is a space, not exactly a familiar kind of Hilbert Space, but a non-separable Hilbert space. Nonetheless, we can now translate from one Hilbert space to the next.

If we call, , where exists in a different space, specifically,

Of which their exists an inner product (This can be explained better with a good understanding of Hilbert spaces; this is just a mapping between spaces). In this new space the inner product looks like,

For an appropriately chosen measure (where again, it requires some work to justify this, but it's something in this vector). This allows us to no longer talk about as a function on its own, but rather talk about other functions which exist in a different Hilbert space, and approximating the solution in the different Hilbert space.

I apologize if I'm being a bit top heavy in my analysis right now. I'm going to start a paper on this, mostly with the goal of finding a different manner of defining uniqueness and expressibility of Kneser's Tetration. Even if I have to use Kneser's tetration to do everything I'm going to do; I think it may be very important to view it as a Hilbert Space, and view the construction in a different Hilbert space.