On which conditions on A$\subseteq\mathbb{R}$ is $\left\|\bullet\right\|:\mathbb{R}[x]\to\left[0,\infty\right]:P\mapsto\sup_{x\in A}\left|P(x)\right|$ Let A$\subseteq\mathbb{R}$.
On which conditions on A$\subseteq\mathbb{R}$ is $\left\|\bullet\right\|:\mathbb{R}[x]\to\left[0,\infty\right]:P\mapsto \sup_{x\in A}\left|P(x)\right|$
I've concluded that there is no need for A to have any conditions for $\left\|\bullet\right\|$ to satisfy:

*

*$\forall P\in \mathbb{R}[x]: \left\| P\right\| =0\Leftrightarrow P=0$

*$\forall P\in \mathbb{R}[x], \forall k\in \mathbb{N}: \left\|kP \right\|=\left| k\right|\left\| P\right\|$
But now I can't find any conditions which would assure that $\forall P,Q \in \mathbb{R}[x]: \left\| P+Q\right\|\leq \left\| P\right\|+\left\| Q\right\|$.
 A: I think $A$ can be arbitrary but non empty.
Let $A \subseteq \mathbb{R}$, such that $A \neq \emptyset$. Let $P(x),Q(x) \in \mathbb{R}[x]$. Notice that by the triangle inequality for all $x \in A$
$$
|P(x)+Q(x)| \leq |P(x)|+|Q(x)| 
$$
Which means that
$$
\sup_{x \in A} |P(x)+Q(x)| \leq \sup_{x \in A} |P(x)+Q(x)|
$$
Then $||P+Q|| \leq ||P||+||Q||$
A: I assume that in (2) you have either established that $0\cdot\infty=0$ or that $0\notin \Bbb N$.

*

*If $A=\varnothing$, then there's the matter of defining what $\sup_{x\in \varnothing}\lvert P(x)\rvert$ is supposed to mean. Usually you want $\sup$ of an empty family to be the minimum of the set of interest, therefore you could make an argument for constant $-\infty$ or for constant $0$. Either way, (2) and (3) hold and (1) doesn't.


*If $A$ is unbounded, then $$\sup_{x\in A}\lvert P(x)\rvert=\begin{cases}\lvert P\rvert &\text{if } P\text{ is constant}\\ \infty&\text{if }P\text{ isn't constant}\end{cases}$$
The function satisfies evidently (2) and (3). It satisfies (1) because $A$ is an infinite set.


*If $A\ne\varnothing$ is bounded, consider the Banach space $(B(A),\lVert\bullet\rVert_\infty)$ of bounded functions on $A$, endowed with the sup norm. Then, your $\lVert \bullet\rVert$ is just the seminorm $p\mapsto\left\lVert \left.p\right\rvert_A\right\rVert_\infty$. Therefore it satisfies (2) and (3), while it satisfies (1) if and only if the restriction map $p\mapsto\left.p\right\rvert_A$ is injective on $\Bbb R[x]$. Id est, if and only if $A$ is also infinite.
