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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?

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    $\begingroup$ As it is currently phrased this looks more like a Mathematics question than a Mathematics Educators question. Can you please explain the educational aspect of this question? Can you please edit that explanation into your question? $\endgroup$ Apr 27 at 0:11
  • $\begingroup$ I would say the motivation for thinking about normed vector spaces really comes from spaces of functions. Many of these are infinite dimensional, but it already looks like a genuine question in the case of the usual dual space of linear functions from a vector space to the underlying field - there are legitimately different notions of how far two linear functions are from each other. $\endgroup$ Apr 27 at 0:19
  • $\begingroup$ The huge difference between normed vector spaces and metric spaces resides mainly in the vector space structure. Given a finite dimensional vector space, one can always manufacture a norm by imposing the usual Cartesian formulas with respect to a given basis. I think the real question here is just what are good real life examples of vector spaces ? Anyway, Ben Crowell's answer is likely what you seek here. $\endgroup$ Apr 27 at 6:25
  • $\begingroup$ You have to be careful when making this kind of "real life" examples. For instance, the function you defined for a mountain terrain may not be a metric, because there are many cases in which the triangle inequality can be easily violated (e.g. if along a shorter path between two points you have to climb a wall). My suggestion is to make examples from applications, as also suggested in Ben Crowell's answer. $\endgroup$ Apr 27 at 7:51
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    $\begingroup$ @StevenGubkin as someone who eats what I like as opposed to satiating the metric of some nutritionist, I suppose I must agree it has no meaning to me. That said, surely there exists a celebrity diet which the length of the vector is the upper bound on the allowed intake of food. Projections into the $x$ or $y$ directions would fall into either the all protein or all carb fad diets. $\endgroup$ Apr 27 at 16:53
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I assume the question is based on your desire to show your students some examples of real-world motivation for the context. If so, then good for you, but it would be nice if you could edit the question to spell that out a little more, including an explanation of the level of the course and your students' background.

[EDIT] The OP edited the question to say that this is for a young kid starting algebra, not a class of college students, so a lot of the examples below will be at too high a level.

Every type of vector you encounter in freshman physics lives in some normed vector space. Force vectors are a normed vector space, as are momenta, velocities, displacements, magnetic fields, and so on. These examples historically were the ones that led to the creation of the vector-scalar system by Gibbs and Heaviside around 1880, so they are the prototypical examples of the word "vector."

For infinite-dimensional examples, I think the earliest important examples were also from physics -- wavefunctions in quantum mechanics. In this application, the norm has an interpretation relating to probability. This norm arises from a complex-valued inner product, which is a measure of the similarity of one wavefunction to another.

Other types of waves, such as sound waves, also live in infinite-dimensional vector spaces, and there the norm typically is a measure of energy. One can define a real-valued inner product between sound waves and such, but it's not particularly physically interesting to my knowledge.

It's actually not easy to find a student-friendly physics example of a vector space that is not normed. Relativity uses vectors with what physicists call a metric, but it isn't positive-definite. So that's a good non-example, but most students probably won't have the background knowledge to be able to understand it. One of the foundational problems in attempts to create a theory of quantum gravity is that it doesn't seem possible to define a norm, and yet we can't make sense of quantum mechanics without a norm. Maybe a somewhat less far-out example of a non-normed vector space in physics would be that sometimes we want to talk about idealized waves such as plane waves, whose norm would diverge.

From outside the field of physics, the best examples I know of are from statistics, where things like correlations can be expressed as inner products, and again norms are probabilities.

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    $\begingroup$ Also significant, I think, is the formalization in terms of normed spaces much of the earlier work in the Calculus of Variations from the 17th and 18th centuries, which as I'm sure you know, is extensively used throughout physics. $\endgroup$ Apr 27 at 6:22

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