Question

Show that if \(\|\cdot\|\) is any norm on \(\mathbb{R}^{n}\) then there exist constants \(a\) and \(b\) such that, $$ a\|x\| \leq\|x\|_{\infty} \leq b\|x\|, \quad \text { for all } x \in \mathbb{R}^{n} $$

Short answer

Question: Prove that for any norm \(\|\cdot\|\) on \(\mathbb{R}^{n}\), there exist constants \(a\) and \(b\) such that the inequality \(a\|x\| \leq\|x\|_{\infty} \leq b\|x\|\) holds for all \(x \in \mathbb{R}^{n}\). Answer: We first analyze the infinity norm for the unit vectors in \(\mathbb{R}^n\), and denote the minimum and maximum of these norms as \(a\) and \(b\), respectively. We then show that for any vector \(x \in \mathbb{R}^n\), the desired inequality holds as \(a\|x\| \leq b\|x\|_{\infty}\) and \(\|x\|_{\infty} \leq a\|x\|\), proving the existence of constants \(a\) and \(b\).

Step-by-step solution

Determine bounds on the infinity norm for the unit vectors

To find constants \(a\) and \(b\), let's first take a closer look at the infinity norm values for the unit vectors \(\{e_1, e_2, .., e_n\}\) in \(\mathbb{R}^n\), where \(e_i\) is the vector with a \(1\) in the \(i^{th}\) position and \(0\) elsewhere. For each unit vector \(e_i\) (\(1\leq i\leq n\)), we have: $$ \|e_i\|_{\infty} = \max_{1\leq j\leq n} |e_{i_j}| = 1. $$ Now, since \(\|\cdot\|\) is a norm, we have \(\|e_i\| > 0\) for each \(i\). Let's denote the minimum and maximum of these norms as $$ a=\min_{1\leq i\leq n} \|e_i\| \quad \text{and} \quad b=\max_{1\leq i\leq n} \|e_i\|. $$ Then, for all \(1\leq i\leq n\), we have \(a \leq\|e_i\| \leq b\).

Analyze the inequality with respect to the infinity norm

Now, let's take any vector \(x \in \mathbb{R}^n\). We want to show that $$ a\|x\| \leq\|x\|_{\infty} \leq b\|x\|. $$ For the left inequality, using the triangle inequality for the given norm, we have $$ a\|x\| = a\left\lVert\sum_{i=1}^n x_ie_i\right\rVert \leq a\left(\sum_{i=1}^n |x_i|\|e_i\|\right) \leq a\left(\sum_{i=1}^n |x_i|(B)\right) = b\sum_{i=1}^n |x_i| \geq b\max_{1\leq i\leq n} |x_i| = b\|x\|_{\infty}. $$ Thus, we have $$ a\|x\| \leq b\|x\|_{\infty}. $$ For the right inequality, let \(m=\text{arg} \max_{1\leq j\leq n} |x_j|\). Then, using the same property as before, we have $$ \|x\|_{\infty} = |x_m| \leq \sum_{i=1}^n |x_i| \leq \sum_{i=1}^n |x_i|\|e_i\| \leq \sum_{i=1}^n |x_i|(A) = a\sum_{i=1}^n |x_i| \leq a\left\lVert\sum_{i=1}^n x_ie_i\right\rVert = a\|x\|. $$ Thus, the desired inequality holds for all \(x \in \mathbb{R}^n\): $$ a\|x\| \leq\|x\|_{\infty} \leq b\|x\|. $$

Key concepts

Infinity Norm

The infinity norm, also known as the "max norm," is a way to measure the size or length of a vector in \( \mathbb{R}^n \). It is denoted as \( \|x\|_{\infty} \). The infinity norm of a vector is the maximum absolute value of its components. In mathematical terms, for a vector \( x = (x_1, x_2, \ldots, x_n) \), the infinity norm is defined as: \[ \|x\|_{\infty} = \max_{1 \leq i \leq n} |x_i|. \] This means, among all the components of the vector \( x \), we find the one with the largest absolute value. This value becomes the infinity norm of \( x \). In practice:

• Useful for assessing the dominant impact of any one element in the vector.
• Widely used in optimization and numerical analysis, especially when dealing with bounding vectors.
• Simplifies the comparison of vectors by focusing on the largest deviation.

An important property of the infinity norm is that for unit vectors (vectors with one element as 1 and all others as 0), the infinity norm is always 1.

Triangle Inequality

The triangle inequality is an essential property of norms in mathematics, including the infinity norm. It basically states that the sum of the lengths of any two sides of a triangle is greater than or equal to the length of the third side.In terms of vector norms, the triangle inequality can be formulated as: \[ \|x + y\| \leq \|x\| + \|y\|, \] where \( x \) and \( y \) are any vectors from a vector space. This inequality ensures:

• That the direct distance between two points (vectors) is at most equal to the sum of lengths of any other two paths connecting the same points.
• The idea behind the triangle inequality helps establish the validity of different types of norms, providing a structure for analyzing vector lengths.
• It is utilized in proving theoretical results, such as convergence properties and stability in numerical methods.

For the infinity norm, this property assists in demonstrating inequalities that hold between different norms on vector spaces, like showing that \( a\|x\| \leq \|x\|_{\infty} \leq b\|x\| \).

Unit Vectors

Unit vectors play a central role in understanding vector norms. A unit vector is a vector with a norm of 1. They serve as a standard basis in vector spaces and are crucial in various applications such as direction measurement, simplifying vector equations, and forming a complete basis for representing other vectors. In \( \mathbb{R}^n \), unit vectors are typically represented as \( e_i \), where \( e_i \) is the vector with 1 in its \( i^{th} \) position and 0 elsewhere. For example:

• In \( \mathbb{R}^2 \), the unit vectors are \( e_1 = (1, 0) \) and \( e_2 = (0, 1) \).
• In \( \mathbb{R}^3 \), they are \( e_1 = (1, 0, 0) \), \( e_2 = (0, 1, 0) \), and \( e_3 = (0, 0, 1) \).

The importance of unit vectors lies in their ability to facilitate the calculation of vector components and their norms. With the infinity norm, a unit vector always has a norm of 1, which is why they are often used in analyses such as finding constants \( a \) and \( b \) that bound the infinity norm. They help define boundaries and explore the properties of vector spaces systematically.

Vector Spaces

Vector spaces provide the foundational framework for the study of vectors and vector norms. A vector space is a collection of vectors, where vectors can be added together and multiplied by scalars to produce another vector within the same space. This concept encompasses a variety of mathematical disciplines, from linear algebra to functional analysis. Key features of vector spaces include:

• Closure under addition and scalar multiplication, which means combining vectors still results in a vector from the same space.
• Existence of a zero vector (a vector with all components zero), which acts as an identity element for addition.
• Inherently, every vector can be represented as a linear combination of a set of basis vectors, such as the unit vectors discussed earlier.

Understanding vector spaces is crucial when dealing with different types of vector norms, including the infinity norm. In any given vector space, norms provide a way to measure vectors' "size" or "length," offering a standardized way to compare and analyze them. With norms and spaces, concepts like the triangle inequality help ensure such measurements are consistent and mathematically sound.