Definition talk:Projection (Hilbert Spaces)

This definition seems to be overly restrictive. Any definition of a projection that I've come across is that it be an idempotent linear operator on a vector space; the definition on this page suggests that all projections are orthogonal projections (i.e. self-adjoint projections), which is obviously not the case. Nm420 17:01, 22 April 2012 (EDT)

Interesting. If you can find a citation for this, please post it up. One of our features of this site is that we attempt to document variants in definitions. --prime mover 18:14, 22 April 2012 (EDT)

At the very least, Halmos (in "Finite-dimensional vector spaces") defines a projection as an idempotent linear operator; I'm sure I've seen this definition in other texts/monographs as well, though nothing else specific comes to mind at the moment. While I could post an "alternative" definition, it seems there should just be two separate definition pages: one for a projection (an idempotent linear operator), and one for an orthogonal projection (a self-adjoint projector, which then implies the kernel and range are orthogonal). I'm hesitant to make any serious changes, though, as there are quite a few Theorem pages which assume that a projection is what I would call an orthogonal projection. Nm420 18:42, 22 April 2012 (EDT)

What you call a projection is covered as an Definition:Idempotent Operator. I deem the name 'projection' in that case a tad misleading since eg $e_1,e_2\mapsto e_1$ is idempotent but not really (intuitively) a projection (in $\R^2$). Note that I will be covering a lot of the pages in the FA domain in the future again, following Conway and generalising them from Hilbert space to Banach space, topological vector space and locally convex space as those apply. Lecture notes I looked up also defined projections self-adjoint (but these courses were closely tied to Hilbert spaces and C* algebras).

I agree that the terminology 'Projection' in your sense should be covered on PW, and I think it's best put as a disambiguation/redirect combination to Idempotent Operator (which name is IMO more descriptive). Does that sound like a suitable solution? --Lord_Farin 11:06, 23 April 2012 (EDT)

That does sound quite acceptable. Nm420 11:54, 23 April 2012 (EDT)

Put it up on Definition:Projection, and also added some about templates for ease of navigation. Thanks for introducing this convention on PW. --Lord_Farin 13:33, 23 April 2012 (EDT)

Looks beautiful. Thanks! Nm420 16:02, 23 April 2012 (EDT)