locally nonsatiated preferences

what does this symbol mean in the discuss of locally nonsatiated preferences: $$\varepsilon > 0$$ and $$||y-x||<\varepsilon$$.

If you take the vector $$\mathbf{x}$$ as the center and an $$\epsilon > 0$$ sufficiently small as the radius, then the open ball (local neighborhood of the vector $$\mathbf{x}$$) is the set of all vectors $$\mathbf{y} \in \mathbb{R}^{n}$$ whose distance to the center $$\mathbf{x}$$ is less than $$\epsilon$$ measured w.r.t. the metric $$\Arrowvert \cdot \Arrowvert$$, i.e., more formally as

$$\mathbf{B}(\mathbf{x},\epsilon) := \big\{\mathbf{y} \in \mathbb{R}^{n}\,\Big\arrowvert\, \Arrowvert \mathbf{y} - \mathbf{x} \Arrowvert < \epsilon \big\}.$$

Notice that measuring a distance by a metric is only possible within a metric space or normed space like Hausdorff or Banach. Topological spaces do not have in general a metric, in this case the above concept must be generalized. Fortunately, all finite dimensional spaces have a canonical norm, i.e., the Euclidean norm.

$$||v||$$ is the norm symbol, it means the length of the vector $$v$$.

• Surely you mean "norm". The norm is a special case of distance-functions or "metrics".
– heh
Commented Feb 14, 2020 at 19:16

Depends on what exactly you mean with "this symbol":

1. $$\varepsilon$$ denotes a real number.
2. $$>$$ stands for the strictly-greater-than relation in the real numbers.
3. $$0$$ is the number zero, i.e., the neutral element with respect to addition.
4. $$x$$ and $$y$$ are consumption bundles, i.e., vectors of the same finite dimension with non-negative real entries.
5. $$-$$ is the minus sign, the inverse of the $$+$$ operation. The operation is performed element-wise for vectors.
6. $$||v||$$ is the norm (i.e., length) of the vector $$v$$. If $$v\in\mathbb{R}^n$$, then $$||v||=(v_1^2+\cdots +v_n^2)^\frac{1}{2}$$.
7. $$<$$ stands for the strictly-less-than relation in the real numbers.