Given:the norm ends up being:

E.g. in ${\mathbb{C}}^2$:

A dip circle, also known as a dip needle or magnetic dip instrument, is a type of scientific instrument used to measure the angle of inclination of the Earth's magnetic field relative to the horizontal plane. This angle is known as the magnetic dip or magnetic inclination. The dip circle typically consists of: 1. **A magnetic needle:** This needle is freely pivoted and can rotate in a horizontal plane. The needle aligns itself with the local magnetic field.

${\mathbb{R}}^n$ with extra structure added to make it into a metric space.

The identity matrix.

Each elliptic space can be modelled with a real projective space. The best thing is to just start thinking about the real projective plane.

Generalize function to allow adding some useful things which people wanted to be classical functions but which are not,

It therefore requires you to redefine and reprove all of calculus.

For this reason, most people are tempted to assume that all the hand wavy intuitive arguments undergrad teachers give are true and just move on with life. And they generally are.

One notable example where distributions pop up are the eigenvectors of the position operator in quantum mechanics, which are given by Dirac delta functions, which is most commonly rigorously defined in terms of distribution.

Distributions are also defined in a way that allows you to do calculus on them. Notably, you can define a derivative, and the derivative of the Heaviside step function is the Dirac delta function.

The "0-width" pulse distribution that integrates to a step.

There's not way to describe it as a classical function, making it the most important example of a distribution.

Applications: