Talk:Norm of Hermitian Operator

I know nothing about Hilbert spaces or functional analysis so I have no business in even commenting on this.

But it seems there is a missing something from all this. The way "inner product" is defined, it is just a mapping from $V \times V$ to either $\R$ or $\C$ having certain properties.

The same definition is obviously the same as for a Hilbert space or someone would have said something about it.

So an "inner product" on a Hilbert space is the same as an inner product on any other sort of vector space. Am I right?

But then when you look at Hilbert space, you get this on it: "Make sure to understand the precise definition of (especially) the inner product."

What am I missing? Or do I just not have the necessary mental equipment to be able to understand this? --prime mover (talk) 19:38, 2 August 2021 (UTC)


 * Well saying let $\innerprod \cdot \cdot_\HH$ be an inner product on $\HH$ reads to me like saying let $d$ be a metric on the metric space $X$. In the latter case, we really mean "the metric space $\struct {X, d}$" (or the metric space $X = \struct {Y, d}$ to be picky with reusing symbols) and changing $d$ at all can result in an essentially completely different looking space. $d$ is basically "inherent" to that metric space. Similarly by writing "the Hilbert space $\HH$" we really mean "the Hilbert space $\HH = \tuple {V, \innerprod \cdot \cdot_\HH}$". (the definition page gives a few alternatives, which all get the point across that $\HH$ is a vector space with the topology inherited by (the norm induced by) that inner product)


 * I suppose it's not specifically wrong in some cases, it just seems irregular since we already picked an inner product in defining $\HH$, and any inner product except the one "intrinsic" to $\HH$ won't have anything to do with the topology of $\HH$. It doesn't really matter too much for results like this, but where something topological is concerned (like continuity or convergence) there is potential for confusion. (since "continuous (on $\HH$)" would really mean continuous with respect to the (norm induced by the) inner product "intrinsic" to $\HH$) In particular, if the norms induced by the two inner products aren't equivalent and their topologies are different.


 * We could insist on writing all the tuples explicitly to make it clearer that the inner product is attached to $\HH$. I am not sure what the ominous note there is about. Caliburn (talk) 20:17, 2 August 2021 (UTC)


 * I used to have a dream that every concept was definable and explainable in a simple unambiguous way. Seems not to be possible with Hilbert space, which has a rambling and discursive style along with several different alternative ways of defining it, and no indication of what is essential to the definition and what is merely discursive distraction. Wondering whether to retire. --prime mover (talk) 21:53, 2 August 2021 (UTC)


 * 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.
 * 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)