A little something about the two equivalent definitions of compactness in a metric space

The two statements are equivalent in a metric space: 1. Every infinite sequence has a convergent subsequence. In other words, there is an accumulation point for every infinite sequence. 2. The metric space is convergent. Proving (2) from (1) follows the standard practice of deducing that any open set containing the accumulation point contains allContinue reading “A little something about the two equivalent definitions of compactness in a metric space”

A continuation of the previous post

Let be an n-dimensional vector space, and let be a surjective linear mapping. If is m-dimensional, then is a matrix of order . Is it possible that ? Let us assume that it is. Then has a basis , all of which are mapped to by distinct vectors in . Moreover, these vectors have toContinue reading “A continuation of the previous post”

Dual spaces- An exposition

Studying Dual Spaces can be confusing. I know it was for me. I’m going to try to break down the arguments into a more coherent whole. I am following Serge Lang’s “Linear Algebra” (the Master is my favourite author). However, I am not strictly following the order he follows to develop the theory. Say weContinue reading “Dual spaces- An exposition”