In Stefan Czerwik's "Functional Equations and Inequalities in Several Variables" there are a few lines I don't understand:
Let X be a real linear space and A be a subset of X.
A point a ∈ A is said to be algebraically interior to A iff for every x ∈ X there exists an ε > 0 such that:
λx + a ∈ A for all λ ∈ (-ε,ε).
Can someone explain how this definition is arrived at?
I also don't get the role of ε. Can you make it clear to me?
On
The statement is saying that your element is interior if when you move a small distance away from that element (in the direction of $x$), you are still in the set.
It doesn't actually matter what $\epsilon$ is, it is just a sufficiently small number which will vary depending on what's been "picked" as $x \in X$.
This simply means that $a$ is an interior point of $A$ if, when you move in any direction (the direction is given by $x$) starting from $a$, it will take a certain time (given by $\varepsilon$) until you leave $A$.
For instance, $(1,0)$ is not an interior point of $A=\{(x,y)\in\mathbb{R}^2\,|\,x^2+y^2\leqslant 1\}$, because if you take, say, $x=(1,1)$, then there is no $\varepsilon>0$ such that $(1,0)+\varepsilon(1,1)\in A$.