Norms and topologies on function spaces
(students can come up with $L^1$, $L^\infty$, sometimes $L^2$, and
 pointwise by themselves, really!)
$L^1$, $L^\infty$, pointwise -- rather naturally motivated
  (the former 2 are "the size" of a function, while pointwise is a
  pretty natural notion of convergence)

$L^2$ is nicely motivated as $\chi^2$, i.e., Euclidean distance!!
  (this is great: it ties together so much!)

$L^p$ is not easily motivated -- it's just a natural generalization of
$L^1$ and $L^2$

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the correct topology for Taylor series (compact open, i.e., uniformly
on compact sets) is tricky to motivate -- note in particular that it's
not a norm.

They way I present it is:
(I really like this presentation:
 it makes a pretty abstract notion very down-to-earth)
- coarsely (I don't use this word: I say "zoom way out"),
  [careful what I mean by "coarsely", if I want to be formal
   in the sense of coarse geometry]
  sine looks like a constant 0 function,
  while a polynomial looks like its leading term (e.g.,
  $\frac{1}{2}x^3-3x$ is coarsely $\frac{1}{2}x^3$)
  (I play a bit with this: coarsely a sphere is a point; a torus is a
  point. every compact manifold is...a point.)
  ...so in the -uniform- metric, Taylor series stay very far away from
  functions
- ...but if you "zoom in" to a -fixed- interval (e.g., a window on
  your calculator), -then- the taylor series converges
  -uniformly-
  -on that interval-
