Definition talk:Hilbert Space

Different definition of Hilbert space

A work under study 1975: Bert Mendelson: Introduction to Topology (3rd ed.) defines a Hilbert space as $H = \struct {\R^\infty, d}$ where:

$\ds \R^\infty := \prod_{n \mathop = 1}^\infty \R$

$\ds \map d {u, v} := \paren {\sum_{n \mathop = 1}^\infty \paren {u_i - v_i}^2}^{1/2}$

where $u = \tuple {u_1, u_2, \ldots}, v = \tuple {u_1, u_2, \ldots}$ for $u_1, u_2, \ldots, v_1, v_2 \ldots \in \R$.

Can this be reconciled with the definition as given in this page? Clearly Mendelson's definition is a specific instance of what is given here, in the limited context of a generalization of a Euclidean vector space, but I am having difficulty finding sources to back up Mendelson's definition.

Is anyone able to help? In the meantime I am skipping this chapter in Mendelson until I get an idea of where this is going to need to go. --prime mover (talk) 12:59, 4 January 2015 (UTC)

The space defined above is intended to be the Hilbert sequence space $\map {\ell^2} \R$. However, it omits the square-summability condition which is necessary to make $d$ a metric. Does that provide you with a start for further inquiry? — Lord_Farin (talk) 13:54, 4 January 2015 (UTC)

Thanks -- yes I wondered about that as well ... but as you say, it doesn't have that extra bit.

I may merely add it as an extension of the Euclidean vector space and call it an "infinite-dimensional Euclidean vector space" and take it from there, adding an Also known as. --prime mover (talk) 18:00, 4 January 2015 (UTC)

I think Hilbert space is $\struct {V, d}$, not $V$. So we need modification.--Bltzmnn.k (talk) 04:04, 26 January 2018 (EST)

I am afraid I don't understand your remark. Could you please clarify? — Lord_Farin (talk) 12:15, 28 January 2018 (EST)

Improving Definition Page

I think what is missing are some simple definitions and theorems that if existed then the definition wouldn't be trying to lift everything from below up to the level of a Hilbert Space. Rather than doing this, have some basic definitions and simple theorems to refer to when you need the topology, metric, norm, etc.

There are some statements that I don't believe belong in the definition:

The Hilbert space $V$ may be considered as one of the following:

The complete inner product space $\struct {V, \innerprod \cdot \cdot_V}$ - can be removed since this is a restatement of the definition - nothing is added.

I think this is useful, as it emphasises the importance of $\innerprod \cdot \cdot_V$. --prime mover (talk) 22:19, 3 August 2021 (UTC)

The Banach space $\struct {V, \norm {\,\cdot\,}_V}$ - this is not an equivalent definition, but it is a theorem that Hilbert Space with Inner Product Norm is a Banach Space

The topological space $\struct {V, \tau_d}$ where $\tau_d$ is the topology induced by $d$ - We just need the definition of the inner product topology, and this statement can go.

The vector space $\struct {V, +, \circ}_{\Bbb F}$ - can be removed since it is part of the definition of the inner product space

Make sure to understand the precise definition of (especially) the inner product. - I don't know what this is referring to, but the most likely thing is conjugate symmetry, which may be new if your only other experience with inner products is on $\R$

Furthermore, the parentheses around the argument of linear functionals and linear transformations on $\HH$ are often suppressed for brevity. - This belongs on the pages relating to linear functionals and linear transformations.

Make sure to understand which symbols denote scalars, operators and functionals, respectively. - This should be replaced with a statement on what symbols are used to denote scalars, operators and functionals

The first thing to note is that the only difference between a Hilbert Space and an Innerproduct Space is Completeness. So I would start by adding some definitions for inner product metric and inner product topology. Some simple theorems to go along with it, and then the definition is nothing more than what I have here. --Leigh.Samphier (talk) 11:29, 3 August 2021 (UTC)

Perhaps (even if on this page only) we could retain the notation $\HH = \struct {V, \innerprod \cdot \cdot_\HH}$?

At the moment the inner-productness is hidden in the definition on the top line, and the vital nature of this inner product is not well communicated.

Everything else I am on board with. --prime mover (talk) 22:19, 3 August 2021 (UTC)

To add on to the above, I'm going to rework the Hilbert space related definitions (mainly splitting into normed vector spaces [Banach where completeness is required] and inner product spaces [same as before with Hilbert]) in this sort of format: User:Caliburn/s/fa/Definition:Bounded Linear Transformation. Any opinions, or am I good to go for it? I will try to rework this definition in the process. I have been working with the notation $\struct {\HH, \innerprod \cdot \cdot_\HH}$ fwiw. Caliburn (talk) 19:44, 21 August 2021 (UTC)

I think that would work -- but bear in mind I'm not familiar with this area so I should not be the final arbiter. --prime mover (talk) 20:34, 21 August 2021 (UTC)