I'm trying to explain some basic concept to my kid (he just started learning basic algebra following Discovering Algebra: An Investigative Approach by Jerald Murdock ). For example homeomorphism means a donut is (in a way) the same as a coffee cup.
Coming to metric spaces, it's easy to explain. The basic concept is that there's a distance between any two points. This can be shown as
- taxi distance vs Euclidean distance (which I coined as "pigeon distance")
- a hiking map measuring distances by how many minutes it will take, due to different types of landforms: sand, swamp, clay, meadow, etc
- given two cities on earth, the distance in between is the same but looks quite different in different map projection, so conversely, on the same map, different distances could be defined
- a sphere shape lantern with an ant wandering on the surface, then projected to floor, ceiling and walls of the room, measure the distance of the internal part of the room by the distance on the surface of the lantern.
However I couldn't find simple examples in real life for normed vector space.
I thought of a flat area with a strong wind where walking vertical to the wind is different to walking along the direction of the wind. However the effort is also due to if one walks along the wind direction or against it, so it's not a nice example.
Can you suggest some good real life examples of normed vector spaces?