The image defines the concept of a metric on a vectorial space $E$ over the set of real numbers $\mathbb{R}$.

Here's a breakdown of the content:

1. Metric Definition:
   A metric $d$ on a set $E$ is a function $d: E \times E \rightarrow \mathbb{R}^+$, which maps any pair of elements $(x, y)$ from $E$ to a non-negative real number (i.e., the distance between $x$ and $y$).

   A metric must satisfy the following properties for all elements $x, y, z \in E$:

   • Non-negativity & Identity of Indiscernibles:
     $d(x, y) \geq 0$ and $d(x, y) = 0 \iff x = y$.
     This ensures that the distance between any two points is non-negative and is zero if and only if the points are identical.

   • Symmetry:
     $d(x, y) = d(y, x)$.
     This means that the distance from $x$ to $y$ is the same as from $y$ to $x$.

   • Triangle Inequality:
     $d(x, z) \leq d(x, y) + d(y, z)$.
     This ensures that the direct distance between two points is always less than or equal to any indirect path involving a third point.

2. Metric Space:
   A metric space is the pair $(E, d)$, where $E$ is the set and $d$ is a metric on $E$. Essentially, it is a set equipped with a notion of distance (via the metric).

This is a fundamental concept in mathematical analysis and topology, where the structure provided by the metric allows for the study of continuity, convergence, and geometric properties of spaces.

Do you need further details or have any questions? Here are some related concepts you might want to explore:

1. What are some examples of metrics on $\mathbb{R}$ or other sets?
2. How do we define a norm, and how does it relate to a metric?
3. What are the differences between Euclidean space and other metric spaces?
4. How does the triangle inequality help in proving properties like convergence in metric spaces?
5. What is an open set in a metric space?

Tip: A useful property of metric spaces is that they allow for the definition of limits and continuity, making them foundational in calculus and real analysis.