Talk:Orthogonal complement

Vacuously true?

"In infinite-dimensional Hilbert spaces, it is of some interest to observe that every orthogonal complement is closed in the metric topology—a statement that is vacuously true in the finite-dimensional case."

Why is this vacuously true? For instance, in R⁴, isn't the orthogonal complement of the vector subspace generated by (1,0,0,0) and (0,1,0,0) the vector subspace generated by (0,0,1,0) and (0,0,0,1), which is closed in the metric topology? It's a little trivial since all vector subspaces are closed in the finite case, but it's not vacuously true, which would imply there are no orthogonal complements.--Syd Henderson 05:49, 16 February 2007 (UTC)

You're right, it's not vacuously true (although it can be reformulated so that it is). Anyway, I've reworded the sentence to make it more general, and removed the "vacuously true" claim. --Zundark 13:38, 16 February 2007 (UTC)

Second property

I think that the second property listed, namely W intersect perp(W) ={0} may not always be true. In vector spaces over finite fields, we can actually have W=perp(W). see example given in "Coding and Information Theory" by Steven Roman, Springer, Graduate Texys in mathemeatics, #134, (1992), page 200. —Preceding unsigned comment added by 76.24.237.217 (talk) 15:18, 3 July 2010 (UTC)

The article is talking about inner product spaces, which by definition are over R or C. --Zundark (talk) 15:56, 3 July 2010 (UTC)

Example on special relativity

I am not comfortable with the wording in the example, since it confuses the concepts of an affine space (in this instance Minkowski space) and a vector space. Simultaneity relates to events (points in an affine space, which does not have an origin). Orthogonality relates to vectors in the tangent space. In an encyclopedia, the distinction between the two types of space should be kept clear. — Quondum 05:16, 30 November 2012 (UTC)

On second thoughts, introducing the concept of an affine space at all in this article is inappropriate. Hence all references to Minkowski space, world lines and events should be removed. The example will have to be rephrased in terms of the vector space with an indefinite symmetric bilinear form (which could be called pseudo-Riemannian, Lorentzian or whatever). — Quondum 07:22, 30 November 2012 (UTC)

Reflexive Banach spaces

It is unclear to me what the function of i in the last LaTeX'd equation is: i\overbar{W} = W^{\perp \perp}

It is an injective isomorphism, so should this be i( \overbar{W} ), maybe even with an isomorphic equality symbol? --Lunakillah (talk) 06:14, 10 February 2014 (UTC)

Finite dimenstions

In the section about finite dimensions, there's a claim about the relation between the row space and the null space of a matrix. It is claimed that they are perpendicular. This statement is only true if the matrix is real. Peleg (talk) 09:17, 15 January 2016 (UTC)