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)