In this article, I will examine various endowments and embellishments which can serve as sources of geometrical structure on a vector space. In a typical first course on linear algebra, even one aimed at students of pure mathematics, the intuition for the notion of a vector space is developed through the use of illustrations decidedly Euclidean in nature. I argue that this is a pedagogical faux pas, mainly for the reason that such pictorial representations give students the misconception that metric structure, or even worse, notions such as projection and angle, are intrinsic to a vector space.
It turns out, in fact, that even something as modest as metric structure is not intrinsic, but rather is induced by something external to the object, namely, a norm. So we often speak of normed linear spaces. Every norm induces a metric, and hence a topology. However, not every topology arises from a metric, of course, and therefore we sometimes concern ourselves with much more abstract objects known as topological vector spaces. Moving in the other direction now, if we have some distinguished non-degenerate bilinear form (in particular, an inner product), then we obtain relatively tame notions of projection, angle, orthogonality and so on. An inner product also gives rise to a norm, however not every norm arises from an inner product. Overall, the hierarchy of “niceness” looks something like this:
My hunch is that our quest to harvest these geometric delectations has a lot to do with the isomorphism (or perhaps even a monomorphism would suffice, with a view towards the infinite-dimensional case). Therefore I want to analyze, in detail, the set of all such isomorphisms. Another interesting point is that an automorphism of is an invertible linear operator. I am led to believe that the more isomorphisms we have, the more such invertible linear operators I can construct, merely by taking distinct isomorphisms and considering maps like and so on. I am just thinking out loud here. Of course, I don’t think every element of arises in this way (think of the infinite-dimensional case). But, recalling that , perhaps there is something to be said here?
Let be a real -dimensional vector space. A non-degenerate bilinear form induces an isomorphism given by . If we replace “bilinear form” with “sesquilinear form” and let be a complex vector space, we obtain something similar, but the isomorphism becomes an anti-isomorphism (we must pay the cost of complex conjugation). Is every isomorphism of this form? Is there some kind of one-to-one correspondence between isomorphisms and a special class of bilinear form?
Note that given any linear functional (“covector”) , we obtain in a natural way an -dimensional subspace — namely, its kernel. This means that if we have a way to associate vectors with covectors, we also have a way to associate -dimensional subspaces with -dimensional subspaces. This is a bijection! Can this be thought of as a crude notion of “orthogonal complement”?
I was originally not going to publish this post yet, but I feel like if I don’t publish it now, I never will. Some of what I said might be wrong, but oh well. I will think about it more and publish more later. This is just a small peek into my thoughts.