Euclidean spaces are multidimensional spaces where each point in the space can be represented by a unique tuple of real numbers that can be called a vector. Addition, the inner product, and the norm of vectors are all defined. Now we can look at inequalities involving vectors.

When we looked at complex numbers, we defined the triangle inequality, |z+w|≤|z|+|w| for any two complex numbers z and w. This is also true for vectors. If **x** and **y** are vectors in a k-space and ||**x**|| is the norm of the vector, then ||**x**+**y**||≤||**x**||+||**y**||. It follows from this that ||**x**−**z**||≤||**x**−**y**||+||**y**−**z**||.

A second related inequality is the Cauchy-Schwarz inequality, which states that |**x**⋅**y**|≤||**x**|| ||**y**||. Essentially, the triangle inequality applies to the addition of vectors and the Cauchy-Schwarz inequality is the same result for the dot product.

A big part of mathematics is taking a specific result and creating a more general rule. One example of that that we've used several times is in extending number systems, starting with the natural numbers, extended up to the real numbers and then complex numbers and euclidean spaces. A second form of that occurred when we looked at the addition and multiplication properties for the rational numbers and declared that these properties define a field, and therefore the rational numbers are a field.

I find wrapping my head around defining fields from the rational numbers significantly harder than understanding defining euclidean spaces from the real numbers. This says to me that the transformation behind the definition of a field is more significant than the transformation behind defining euclidean spaces. And that's a good thing, because it's time for another generalization like the one from rational numbers to fields. Unfortunately, it will have to wait until next time.