Question

Show that \(\\{(s, 0): s \in \mathbb{R}\\}\) is closed in \(\mathbb{R}^{2}\) with any of the norms \(\|\cdot\|_{1},\|\cdot\|_{2}\) or \(\|\cdot\|_{\infty}\)

Short answer

Q: Show that the set \(\{(s, 0): s \in \mathbb{R}\}\) is closed in \(\mathbb{R}^{2}\) with respect to any of the given norms \(\|\cdot\|_{1},\|\cdot\|_{2}\) or \(\|\cdot\|_{\infty}\). A: We have shown that for any sequence \( (x_n, y_n) \) in the set \( S = \{(s, 0): s \in \mathbb{R}\} \), the limit point \((x, y)\) also belongs to the set for each of the three norms (\(\|\cdot\|_{1}\), \(\|\cdot\|_{2}\), or \(\|\cdot\|_{\infty}\)). Therefore, the set \(S\) is closed in \(\mathbb{R}^{2}\) with respect to any of the given norms.

Step-by-step solution

Define the set and norms

First, let's define the given set and the norms: - The set is \(S = \{(s, 0): s \in \mathbb{R}\}\). - The norms are given by: - The \(1\)-norm: \(\|x\|_{1} = |x_{1}| + |x_{2}|\) for \(x = (x_{1}, x_{2}) \in \mathbb{R}^2\) - The \(2\)-norm (Euclidean norm): \(\|x\|_{2} = \sqrt{x_{1}^2 + x_{2}^2}\) for \(x = (x_{1}, x_{2}) \in \mathbb{R}^2\) - The \(\infty\)-norm: \(\|x\|_{\infty} = \max \{|x_{1}|, |x_{2}|\}\) for \(x = (x_{1}, x_{2}) \in \mathbb{R}^2\)

Show limit points belong to the set for 1-norm

Let \((x_n, y_n)\) be a sequence in \(S\) such that \((x_n, y_n) \to (x, y)\) as \(n \to \infty\). Then, since \((x_n, y_n) \in S\), we know that \(y_n = 0\) for all \(n\). We have: \(\|(x_n, y_n) - (x, y)\|_{1} = \|(x_n - x, y_n - y)\|_{1} = |x_n - x| + |y_n - y| = |x_n - x| + |0 - y| \to 0\) as \(n \to \infty\) Since \((x_n - x) \to 0\) and \((y_n - y) \to 0\) as \(n \to \infty\), we have \((x, y) \in S\).

Show limit points belong to the set for 2-norm

Again, let \((x_n, y_n)\) be a sequence in \(S\) such that \((x_n, y_n) \to (x, y)\) as \(n \to \infty\). We have: \(\|(x_n, y_n) - (x, y)\|_{2} = \|(x_n - x, y_n - y)\|_{2} = \sqrt{(x_n - x)^2 + (y_n - y)^2} = \sqrt{(x_n - x)^2 + (0 - y)^2} \to 0\) as \(n \to \infty\) Since \((x_n - x)^2 \to 0\) and \((y_n - y)^2 \to 0\) as \(n \to \infty\), we have \((x, y) \in S\).

Show limit points belong to the set for ∞-norm

Once more, let \((x_n, y_n)\) be a sequence in \(S\) such that \((x_n, y_n) \to (x, y)\) as \(n \to \infty\). We have: \(\|(x_n, y_n) - (x, y)\|_{\infty} = \|(x_n - x, y_n - y)\|_{\infty} = \max \{|x_n - x|, |y_n - y|\} = \max \{|x_n - x|, |0 - y|\} \to 0\) as \(n \to \infty\) Since \(\max \{|x_n - x|, |y_n - y|\} \to 0\) as \(n \to \infty\), we have \((x, y) \in S\). Therefore, we have shown that the set \(S = \{(s, 0): s \in \mathbb{R}\}\) is closed in \(\mathbb{R}^{2}\) with any of the norms \(\|\cdot\|_{1},\|\cdot\|_{2}\) or \(\|\cdot\|_{\infty}\).

Key concepts

Normed Spaces

Normed spaces are foundational in understanding and working with various mathematical concepts, particularly in functional analysis and topology. Here, the idea is to assign a length or "size" to each vector in a vector space, using a function called a norm.
**Understanding Norms**
A norm is a function that takes a vector and returns a non-negative number that symbolizes the vector's length. For example, in o **1-norm**, the length is calculated as the sum of the absolute values of the vector components: \(\|x\|_{1} = |x_{1}| + |x_{2}|\).
- **2-norm**, often referred to as the Euclidean norm, is the usual notion of distance in geometry, defined as the square root of the sum of the squares of the vector components: \(\|x\|_{2} = \sqrt{x_{1}^{2} + x_{2}^{2}}\).
- **\(\infty\)-norm**, gives the maximum absolute component value: \(\|x\|_{\infty} = \max\{|x_{1}|, |x_{2}|\}\).
Each of these norms provides different insights and is useful in various contexts of mathematical analysis and geometry. Normed spaces allow the exploration of convergence, continuity, and even geometry in general terms, which are essential concepts in analysis.

Limit Points

In the realm of mathematics, particularly in topology and analysis, a limit point is a point that can be approached by a sequence from within the set. Understanding limit points is crucial for analyzing the properties of sets.
**Definition and Properties**
A point \((x, y)\) is considered a limit point of a set \(S\) if there exists a sequence \((x_n, y_n)\) in \(S\) that converges to \((x, y)\). This means:
- As \(n\) tends towards infinity, \((x_n, y_n)\) gets arbitrarily close to \((x, y)\).
- The distance between \((x, y)\) and \((x_n, y_n)\) with respect to a chosen norm approaches zero.
For example, if you use the 1-norm, you are looking at the convergence \(\|(x_n, y_n) - (x, y)\|_{1} \to 0\). Similarly, for the 2-norm and \(\infty\)-norm, the process is the same but the measure of distance changes according to the norm being used.
Limit points relate closely to the concept of closed sets. A set is closed if it contains all its limit points. The ability to recognize and prove the existence of all possible limit points in a given set is critical for determining its closure properties.

Closed Sets

Closed sets are an important concept in abstract analysis and topology. Understanding what makes a set closed helps in grasping more complex ideas down the line.
**Characteristics of Closed Sets**
A set is defined as closed if it contains all its limit points. This means:
- Any sequence from within the set that converges will have its limit within the same set.
- Complementing it with the idea of open sets, closed sets contain their boundary points.
**Proving a Set is Closed**
To demonstrate that a set is closed in a space equipped with a norm, like \(\mathbb{R}^{2}\) with norms \(\|\cdot\|_{1},\|\cdot\|_{2}\), or \(\|\cdot\|_{\infty}\), the typical approach is proof through sequences.
- Take any sequence in the set and show that if it converges to some point, that point must belong to the set.
- This means for the set \(\{(s, 0): s \in \mathbb{R}\}\), any sequence of points in this set, under any of these norms, will converge to another point in the same set, making it closed.Closed sets, thus, create a sort of certainty that limit behavior is contained within the set itself, making them highly useful in analysis where "closure" is synonymous with stability in terms of sequences and limits.