Space(s)

“Space, the final frontier”. Growing up, these were the words that kicked off my favorite event of the week - eating pizza and watching Star Trek on Friday nights with my dad and brother. I don’t know why some people are attracted to science, and its associated dreamy holodeck, science fiction, while others wonder what all the fuss is about. For me, it was a blend of seeing compassionate, principled rationalism modeled by a diverse interspecies cast that included “artificial” life and beings from other planets, and the general spirit of adventure that infused each episode of the Next Generation arc.

But it was also the introduction to space-time concepts that pepper the entire Star Trek franchise. Worm holes. Warp speed. Cloaking devices. These fan the imagination, and plants seeds for a life of subterranan inquiry.

On such thread for me has been space itself. What constitutes a space? Can familiarity with mathematical spaces beyond Euclidean space rendered with Cartesian coordinates spur generative imagination?

Preliminaries

Coordiante systems

Cartesian coordinates

Polar coordinates

Elliptical coordinates

Complex space

Linear and Topological Spaces

Euclidean Space

Affine and projective spaces

Function Space

Vector Space

Metric and Uniform Spaces

Normed, Banach, and Hilbert spaces

Hilbert Space

Banach Space

Smooth and Riemannian manifold spaces

Measure and probability spaces

Complete normed vector space