Stephen Abbott
- Understanding Analysis
There needs to be significant reward for the difficult work of firming up the logical structure of limits. Specifically, real analysis should not be just an elaborate reworking of standard introductory calculus. Students should be exposed to the tantalizing complexities of the real line, to the subtleties of different flavors of convergence, and to the intellectual delights hidden in the paradoxes of the infinite.
I.N.Herstein
- from Topics in Algebra(shortened slightly)
Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. For this reason the basic concepts introduced in them have a certain universality and are ones we encounter, and keep encountering, in so many diverse contexts. Among these fundamental notions are those of linear dependence, basis, and dimension which will be developed in this chapter. These are potent tools in all branches of mathematics.
