##
A Proof of Liouville’s Theorem in complex analysis
*August 31, 2009*

*Posted by Phi. Isett in Uncategorized.*

Tags: complex analysis, Liouville's theorem

2 comments

Tags: complex analysis, Liouville's theorem

2 comments

Liouville’s theorem in complex analysis says that the only bounded holomorphic functions are constant. This is my favorite

**Proof:**

Pretend you had a non-constant, bounded holomorphic function on all of . Since is bounded at , Riemann’s theorem on removable singularities implies that extends to a holomorphic (and hence continuous) function on the Riemann sphere , which is compact. If were not constant, the open mapping theorem would apply to this extension, and the image of the Riemann sphere would be an open subset of . But this cannot be the case, because has no nonempty subsets which are both open and compact.

There are some downsides to this proof. It does not rely on the theory of Riemann surfaces (not a downside). It does, however, rely on some relatively (though not truly) heavy machinery, and in order to be a correct proof, one isn’t allowed to use Liouville’s theorem to develop this machinery (but it is possible). Liouville’s theorem follows rather immediately from Cauchy’s integral formulae, and I don’t personally know how to establish things like analyticity, Riemann’s theorem and the open mapping theorem without this tool (although I would be interested if anyone else does!). I also don’t think it really generalizes very well to other PDE for which a Liouville theorem holds.

##
On the geometric meaning of the Cauchy Schwarz inequality, an intro to exterior powers, and surface integrals
*August 19, 2009*

*Posted by Phi. Isett in Uncategorized.*

Tags: Cauchy Schwarz, Cauchy Schwarz inequality, Determinants, Exterior Algebra, Exterior power, inner product, Surface integrals, Surface integration

3 comments

Tags: Cauchy Schwarz, Cauchy Schwarz inequality, Determinants, Exterior Algebra, Exterior power, inner product, Surface integrals, Surface integration

3 comments

This is a brief remark on the Cauchy Schwarz inequality and one way of understanding its geometric meaning (at least in the context of a real inner product space). In a real inner product space , the inner product allows for a generalization of intuitive geometric notions of “length”, “angle”, and “perpendicular” for vectors in . For two elements , I will write to indicate the parallelogram formed by taking and as edges. The reader may as well simply imagine for his favorite and that the inner product is the usual dot product.

The Cauchy Schwarz inequality says that the area of a parallelogram is positive unless u and v are co-linear (it is also equivalent to the triangle inequality, but I will be talking about this formulation instead). If and are co-linear (they point in the same direction or perhaps opposite directions; maybe ) the parallelogram one forms with these two vectors is degenerate and has zero area; otherwise, you wind up with a parallelogram which has positive area. In fact, the volume of is when the two vectors and are perpendicular. If and fail to be perpendicular, then we can observe that shifting the edge by any amount in the direction of the other edge does not change the area of the parallelogram; thus the area of is the same as the area for any . By choosing to minimize the length of the first edge (pick ), we can make both edges perpendicular. The area of the resulting parallelogram must be non-negative (and must be positive when and are linearly independent), giving the inequality . (This is actually a self-contained proof of the Cauchy-Schwartz inequality, and it’s the usual proof, but with some motivation regarding how to choose .)

Of course, the Cauchy-Schwartz inequality is also equivalent to the triangle inequality, and the relationship between these two geometric interpretations can be seen by inspecting the parallelogram one forms by drawing ““.

I hope to write on how formalizing this area-of-parallelogram concept works (only on an intuitive level, for now). For those already familiar with exterior powers of vector spaces and exterior algebras (and their geometric meaning in terms of parallelograms), I can cut to the chase and say that there’s an induced inner product on the exterior powers and the Cauchy Schwartz inequality is what you get by writing .

I will also discuss the relation to surface integrals and determinants.