Say I'm in continuum mechanics and I'm looking at displacements $u$ of an object $\Omega$; let that object resemble something physical, having dimensions given in, say, meters.
I could now define a reference length, e.g. 1 meter, and express $u$ as a vector field of dimensionless numbers, relative to the reference length; and do the same with velocities (e.g. relative to 1 meter/second).
If I do that, I can all of a sudden add displacements and velocities, because they're both just numbers; by accidient. I'd like to avoid that and stick to dimensionful quantities.
So I'd have a displacement vector field, at each point a vector with components which are dimensionful numbers.
Now I have the following problem (you always have it, but this way you see it):
Take a space like $C^1(\Omega)$ or $H^1(\Omega)$ and equip it with a norm. That norm will most likely be of the form $\|u\| = F(|u|_1,|Du|_2)$ (here, $Du$ would be the (weak) derivative) of $u$) with half-norms $|\cdot|_1$ and $|\cdot|_2$. $F$ could add its two arguments, take a q-sum, the maximum, or something like that -- it doesn't matter so much since everything ends up yielding equivalent norms -- with the goal of making the following statement true:
If the $\|u_n\|$ goes to zero, then so do $|u_n|_1$ and $|u_n|_2$.
Now take dimensions into account.
The problem is: $u$ and $Du$ carry different dimensions (e.g. meters and none). So you cannot add them. And you should not be able to. Because one is about size and the other is about variation. You can circumvent this problem by taking a weighted sum instead, where the weights carry units. But that's is absurd: this way, you could have your norm have any dimension you desire!
What is a way out of this? How can you define a meaningful norm on $H^1(\Omega)$ for dimensionful quantities?