Constructing seminorm from norm Two quick motivating examples:
Example 1: $\mathbb{R}^n$ can be considered as a function space of functions from $n$ to $\mathbb{R}$. Function $p(x_1, \cdots, x_n) = (x_1^2 + \cdots + x_n^2)^{1/2}$ defines a norm if $k = n$, and $p'(x_1, \cdots, x_n) = (x_1^2 + \cdots + x_k^2)^{1/2}$ defines a seminorm for any $1 \leq k < n$.
Example 2: For $f \in L^1[0,1]$, function $r(f) = \int_0^1 |f|$ defines a norm, and $r'(f) = \int_0^k |f|$ defines a seminorm for any $0 < k < 1$.
So it seems that given a norm $q$ on function space $X$ where each $f \in X$ has domain $\Omega$, one can define a seminorm $q'$ by asking that $q'(f)$ "check on" the behavior of $f$ only on a subdomain $\Omega' \subset \Omega$ and ignore the behavior of $f$ on $\Omega \smallsetminus \Omega'$. For example, $p'(x)$ only asks what the first $k$-many coordinates of $x$ are doing, and $r'(f)$ only considers the behavior of $f$ on the interval $[0,k]$. My question is: Is there some terminology associated with this phenomon? Some set of keywords I should look up? As you can see, saying that the seminorm "checks on" the behavior of $f$ on some subdomain $\Omega'$ sounds stupid and unrigorous!
Edit for clarity: What I'm asking is, in these examples, we're not restricting the domain of the norm/seminorm, e.g. norms $p$ and $p'$ have the same domain. But we are in some way restricting the domain of the functions on which the norm acts. Does that object, `the domain of the functions on which the norm acts' have a name?
 A: We can describe these examples by composing with a suitable linear operator. It's easier to describe the $L^1[0, 1]$ example first: we have
$$\int_0^k |f(x)| \, dx = \int_0^1 |f(x) 1_{[0, k]}(x)| \, dx$$
where $1_X$ denotes the indicator function of a subset $X$, which is equal to $1$ on $X$ and $0$ otherwise. We can describe the first example in the same way by thinking of $\mathbb{R}^n$ as the space of functions $\{ 1, 2, \dots n \} \to \mathbb{R}$ and multiplying such a function by the indicator function $1_{[1, k]}(x)$ (which amounts to setting all coordinates except the first $k$ coordinates to zero).
The general pattern is that if $(W, \| \|_W)$ is a normed vector space and $f : V \to W$ is a linear map (where $V$ is just a bare vector space) then the composite $\| v \|_V = \| f(v) \|_W$ is a seminorm on $V$ whose kernel is $\text{ker}(f)$. In this special case $V = W$ is a space of functions and $f$ is an idempotent multiplication operator which sets a function equal to zero outside of a subset. Operators of this form show up, for example, in the spectral theorem.
Alternatively we can take $f$ to be a restriction map between two different function spaces which is maybe a little cleaner, e.g. the restriction map $L^1[0, 1] \to L^1[0, k]$ in the second example and the restriction map $\mathbb{R}^{ \{ 1, \dots n \}} \to \mathbb{R}^{ \{ 1, \dots k \} }$ in the first example. This would allow us to discuss examples like the space of continuous functions $C([0, 1])$ with the sup norm and the restriction maps $C([0, 1]) \to C([0, k])$; here the multiplication-by-an-indicator-function argument is technically not available because $1_{[0, k]}$ is not continuous and multiplication by it does not define an operator on $C([0, 1])$. Thinking about things this way has the sort of funny feature that we pull back twice: once on functions and then again on (semi)norms on functions.
