The definition of a norm

By the end of this lesson, you should be able to:

• Understand the definitions of seminorms and norms in vector spaces over \( \mathbb{R} \) or \( \mathbb{C} \).
• Distinguish between seminorms and norms.
• Explore examples of norms in both finite-dimensional and infinite-dimensional vector spaces.
• Comprehend how norms induce metrics.
• Understand why not every metric space is induced by a norm.
• Apply these concepts to specific spaces like \( \ell^1 \), \( \ell^\infty \), and \( C[a, b] \).

Recall Vector Spaces: A vector space over a field \( F \) (either \( \mathbb{R} \) or \( \mathbb{C} \)) is a set where vector addition and scalar multiplication are defined and satisfy certain axioms. For a detailed review, refer to this resource.

Purpose of the Lesson: We will explore how to measure the “size” or “length” of vectors using norms and seminorms.

Seminorm: A function \( \| \cdot \|: X \rightarrow \mathbb{R} \) on a vector space \( X \) satisfying:

1. Non-negativity: \( 0 \leq \| x \| < \infty \) for all \( x \in X \).
2. Homogeneity: \( \| c x \| = |c| \| x \| \) for all \( c \in F \) and \( x \in X \).
3. Triangle Inequality: \( \| x + y \| \leq \| x \| + \| y \| \) for all \( x, y \in X \).

Norm: A seminorm that also satisfies:

4. Definiteness (Uniqueness): \( \| x \| = 0 \) if and only if \( x = 0 \).

Discussion: Each property ensures that the function \( \| \cdot \| \) behaves like a “length” or “size” measure in \( X \).

Seminorms vs. Norms:

• A seminorm may assign zero length to non-zero vectors.
• A norm assigns zero length only to the zero vector.

Example of a Seminorm that is Not a Norm:

Consider the function \( \| x \| = |x_2| \) on the vector space \( \mathbb{R}^2 \), where \( x = (x_1, x_2) \). This function satisfies the properties of a seminorm:

• Non-negativity: \( \| x \| = |x_2| \geq 0 \).
• Homogeneity: \( \| c x \| = |c x_2| = |c| |x_2| = |c| \| x \| \).
• Triangle Inequality: \( \| x + y \| = |x_2 + y_2| \leq |x_2| + |y_2| = \| x \| + \| y \| \).

However, it is not a norm because it lacks definiteness. There exist non-zero vectors for which \( \| x \| = 0 \):

If \( x = (x_1, 0) \) with \( x_1 \neq 0 \), then \( \| x \| = |0| = 0 \), but \( x \neq 0 \).

Importance of the Definiteness Property: The definiteness property ensures that only the zero vector has zero length, making \( \| \cdot \| \) a true measure of size in a normed vector space.

Define \( F^d \): The \( d \)-dimensional vector space over \( F \).

Examples:

• \( \ell^1 \)-Norm: \( \| x \|_1 = \sum_{i=1}^d |x_i| \) for \( x = (x_1, x_2, \dots, x_d) \in F^d \).
• Euclidean ( \( \ell^2 \) ) Norm: \( \| x \|_2 = \left( \sum_{i=1}^d |x_i|^2 \right)^{1/2} \). Represents the physical length of the vector.
• \( \ell^\infty \)-Norm: \( \| x \|_\infty = \max_{1 \leq i \leq d} |x_i| \).

Visualizing the Triangle Inequality: In \( \mathbb{R}^2 \), the triangle formed by vectors \( x \), \( y \), and \( x + y \) illustrates \( \| x + y \| \leq \| x \| + \| y \| \).

Explore Different p-Norms in 2D and 3D: You can visualize unit balls for different p-norms in both 2D and 3D by visiting this excellent interactive demonstration:
Unit Balls for Different p-Norms in 2D and 3D by Aaron T. Becker and Ravi Patel.
This demonstration is part of the Wolfram Demonstrations Project, published on December 29, 2020.

Introduce Sequence Spaces:

• \( \ell^1 \): Space of sequences \( x = (x_k)_{k \in \mathbb{N}} \) where \( \sum_{k=1}^\infty |x_k| < \infty \). Norm defined as \( \| x \|_1 = \sum_{k=1}^\infty |x_k| \).
• \( \ell^\infty \): Space of bounded sequences. Norm defined as \( \| x \|_\infty = \sup_{k \in \mathbb{N}} |x_k| \).

Discussion: These spaces are essential in functional analysis. Infinite-dimensional norms can behave differently from finite-dimensional ones.

Space \( C[a, b] \): Set of continuous functions \( f: [a, b] \rightarrow F \).

Examples of Norms:

• L1-Norm: \( \| f \|_1 = \int_a^b |f(t)| \, dt \). Homogeneity and Triangle Inequality follow from properties of integrals.
• Uniform Norm: \( \| f \|_u = \sup_{t \in [a, b]} |f(t)| \). Useful for ensuring uniform convergence.

Completeness: \( C[a, b] \) is complete under the uniform norm but not under the L1-norm.

From Norms to Metrics: A norm \( \| \cdot \| \) induces a metric \( d \) via \( d(x, y) = \| x – y \| \).

Examples:

• Finite-Dimensional Spaces: \( d_1(x, y) = \| x – y \|_1 \), etc.
• Infinite-Dimensional Spaces: Similar definitions apply in \( \ell^1 \) and \( \ell^\infty \).

Understanding the Limitation:

While every norm induces a metric through the formula \( d(x, y) = \| x – y \| \), the converse is not true; not every metric arises from a norm. This is due to several reasons:

1. Lack of Vector Space Structure: Some metric spaces are not vector spaces. Norms require operations like vector addition and scalar multiplication, which may not exist in a general metric space.
2. Metrics Not Translation Invariant: For a metric to be induced by a norm, it must satisfy translation invariance:
   \[
   d(x + z, y + z) = d(x, y) \quad \text{for all } x, y, z.
   \]
   Not all metrics have this property.
3. Failure of Homogeneity: Norm-induced metrics are homogeneous with respect to scalar multiplication:
   \[
   d(c x, c y) = |c| \, d(x, y) \quad \text{for all } c \in F.
   \]
   Some metrics do not satisfy this condition.

Example of a Metric Not Induced by a Norm:

Consider the discrete metric on any set \( X \), defined by:
\[
d(x, y) = \begin{cases}
0 & \text{if } x = y, \\
1 & \text{if } x \neq y.
\end{cases}
\]
This metric satisfies all the properties of a metric but cannot be induced by a norm on a vector space because it doesn’t relate to the vector space operations in any meaningful way.

Another example is a modified metric on \( \mathbb{R}^2 \) defined by:
\[
d((x_1, y_1), (x_2, y_2)) = |x_1 – x_2| + 2|y_1 – y_2|.
\]
While \( d \) satisfies the metric properties, it cannot be induced by any norm because it doesn’t scale uniformly with scalar multiplication due to the differing weights.

Summarize Key Points:

• Definitions and differences between norms and seminorms.
• Examples of norms in various vector spaces.
• Relationship between norms and metrics.
• Understanding that not all metrics are induced by norms.