91Ó°ÊÓ

The Euclidean norm is a way to measure the length or size of a vector in the Euclidean space. For any vector \[ \mathbf{x} = (x_1, x_2, ..., x_n) \] in \( \mathbb{R}^n \), the Euclidean norm is defined as: \[ \| \mathbf{x} \| = \sqrt{x_1^2 + x_2^2 + ... + x_n^2} \] This formula essentially follows the Pythagorean theorem to calculate the distance from the origin to the point represented by the vector.

• It's the most natural way of measuring distance in our usual geometric concept of space.
• In the context of normed vector spaces, it's one of the most common and fundamental norms due to its simplicity and empirical origin from physical space measurement.
• The Euclidean norm is often denoted by double vertical bars "\( \| \cdot \| \)", reflecting its significance in indicating vector magnitude.

When discussing open balls in \( \mathbb{R}^n \), the Euclidean norm helps define the concept by determining whether a point is inside or outside the ball based on the inequality \( \| \mathbf{x} - \mathbf{a} \| < r \).

The triangle inequality is a fundamental property of norms, and specifically the Euclidean norm in this case. It states that for any vectors \( \mathbf{u} \), \( \mathbf{v} \) in \( \mathbb{R}^n \): \[ \| \mathbf{u} + \mathbf{v} \| \leq \| \mathbf{u} \| + \| \mathbf{v} \| \] This inequality can be visualized by imagining the scenario in a triangle where the sum of the lengths of any two sides must be greater than or equal to the length of the third side.

• In geometric terms, it means the direct path between two points is the shortest.
• It ensures the distances behave in a consistent way, crucial for the properties of space like those seen in our exercise.
• This inequality simplifies understanding complex vector operations by providing a clear bound on how norms interact during addition.

In the solved exercise, the triangle inequality helps to prove that a neighborhood around a point is contained within the open ball, supporting the proof of openness by controlling the sum of the distances precisely.

In topology, the concept of a neighborhood is pivotal to understanding open sets, like the open ball in Euclidean space. A neighborhood of a point \( \mathbf{p} \) in \( \mathbb{R}^n \) is essentially a set of points surrounding \( \mathbf{p} \) that belongs to the larger set \( S \). Formally, for a point \( \mathbf{p} \) and radius \( \epsilon > 0 \), the neighborhood is defined as: \[ N(\mathbf{p}, \epsilon) = \{ \mathbf{x} \in \mathbb{R}^n : \| \mathbf{x} - \mathbf{p} \| < \epsilon \} \] Having neighborhoods contained within open sets is key in determining openness.

• An open set includes a neighborhood around each of its points.
• This property allows the open ball \( B(\mathbf{a}, r) \) itself to be open when every point \( \mathbf{p} \) has a fully enclosed neighborhood inside the ball.
• The concept helps generalize the understanding of continuity and limit processes within different spaces.

For our exercise, proving that each point in the open ball has a neighborhood also in the ball is crucial. Using the Euclidean norm, we compute these neighborhoods precisely enough to ensure that they remain inside, illustrating the ball's openness.