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

<p><span class="math-container">$$\mathbf{B}(\mathbf{x},\epsilon) := \big\{\mathbf{y} \in \mathbb{R}^{n}\,\Big\arrowvert\, \Arrowvert \mathbf{y} - \mathbf{x} \Arrowvert &lt; \epsilon \big\}.$$</span></p>

<p>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.    </p>

CC BY-SA 4.0
2020-02-14T12:08:49.787Z
2020-02-14T12:14:31.343Z
2020-02-14T12:14:31.343Z
Holger I. Meinhardt
<p>Depends on what exactly you mean with "this symbol":</p>

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

CC BY-SA 4.0
2020-02-19T10:02:05.807Z
2020-02-19T10:02:05.807Z
VARulle
<p><span class="math-container">$||v||$</span> is the <em>norm</em> symbol, it means the length of the vector <span class="math-container">$v$</span>.</p>

CC BY-SA 4.0
2020-02-14T19:16:46.920Z
Surely you mean "norm".  The norm is a special case of distance-functions or "metrics".
heh
CC BY-SA 4.0
2020-02-14T05:54:56.547Z
2020-02-14T20:38:28.220Z
2020-02-14T20:38:28.220Z
Giskard
<p>what does this symbol mean in the discuss of locally nonsatiated preferences:
<span class="math-container">$\varepsilon &gt; 0$</span> and <span class="math-container">$||y-x||&lt;\varepsilon$</span>.</p>

<microeconomics><utility><preferences>
locally nonsatiated preferences
Dennis Ajala
# locally nonsatiated preferences

(Asked by: username_0 on 2020-02-14)

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

**Tags:** microeconomics, utility, preferences


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## 3 answers

### Answer by username_1 on 2020-02-14. Score: 1

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.


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### Answer by username_2 on 2020-02-19. Score: 0

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.


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### Answer by username_3 on 2020-02-14. Score: 0

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

> **Comments:**
> **username_4**: Surely you mean "norm".  The norm is a special case of distance-functions or "metrics".  

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