Recent content by SVN

Is there any use for this concept in classical branches of physics? Can it be of any help for a physicist in resolving problems (or, at least, in resolving them more efficiently when compared with traditional methods)? The word «classical» means exactly that, i. e. mechanics, hydrodynamics...

@fresh_42 What is wrong with regarding ##v^T\cdot v## as an inner product automatically?

@fresh_42 Let me explain myself. Of course, I did not mean the inner product operation and matrix multiplication to be the same. Let's say we have two vectors (vector and covector to be precise) that we will regard as matrices with one row and one column. So, by definition of inner product we...

@fresh_42 I am afraid I am missing your point. We can multiply only those matrices that have equal numbers of rows and columns. For example we can multiply a matrix 2x3 by another matrix 3x4. But how should we multiply 2x3 by 4x4? It is not defined, is it? So, how can one refer to set of...

Well, the question is in the title.

It says in any textbook (for example, in classical text «Theory of matrices» by P. Lankaster) on matrix theory that matrices form an algebra with the following obvious operations: 1) matrix addition; 2) multiplication by the undelying field elements; 3) matrix multiplication. Is the last one...

Well, of course, but I meant its classification from the viewpoint of functional analysis.

Thank you!

I am not sure I understand your point. The analytic functions form a small and restrictive class of functions. It can be broadened by dropping some requirements imposed on class members. It gives us this sequence (incomplete, I guess, but it illustrates the basic idea): ##C^\omega \subset...

Yes, you are right. I was thinking about interval [-1,1].

For historical reasons the hyperbola always was considered to be one of the «classical» curves. The function, obviously, does not belong to C0. Apparently, is does not fit L2 or any other Lp? What is the smallest class?

The way I see it the discussion could now be closed. At least I got my answer and it certainly makes sense to me. Thanks a lot to everyone involved.

It was not about higher dimensions for me, actually. It is just an example. If one talks about 2D tangent space, he uses vectors, not complex numbers. It is probably possible to consider tangent plane as Argand plane, but no one is doing that (AFAIK). There must be a reason for that.

I can't argue with that. But I fail seeing how it answers my question, sorry. I realize the difference results of this extra algebraic structure. My question is how to understand whether this extra is helpful for solving a task (thus choosing between the two formalisms to work with the problem...

Well, let's consider a specific problem (the way I see it is still my original question, just reformulated for specific situation). There is a 2D regular surface in Euclidean 3D space. If one talks about tangent space in any of its point, he necessarily use concept of 2D linear space. Vectors...