Inner Products
An inner product is an additional bit of structure which we can impose on a vector space which allows the definition of things like the "magnitude" (or "norm") of vectors, as well as the "angle" between two vectors.
Eric Shapiro
Constructing the Real Numbers (2) - Cauchy Sequences
In my last post we explored the nature of the gaps in the rational number line. In doing so, we defined Cauchy sequences and discovered that rational Cauchy sequences do not always converge to a rational number!
Constructing the Real Numbers (1)
We are going to follow a similar procedure now and attempt to construct the real numbers $\R$ using the rational numbers $\Q$ as a starting point. This is a bit more challenging, due to the more subtle nature of the gaps between rational numbers, and so it requires some more complex machinery
Compactness (2) - The Heine-Borel Theorem
In this post, I will prove a series of seemingly random conjectures about compactness. However, each of them will be required in order to prove the main result of this post: The Heine-Borel Theorem. This theorem completely characterizes compact sets in euclidean space.
Compactness (1)
Unlike the other topological invariants we've discussed, such as connectedness, compactness does not have an immediately obvious geometric description. However, it is a concept that is impossible to avoid, as it shows up everywhere and in every field of mathematics.