Important Formulas for Review and Metric Spaces

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.
• 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.

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: \( \| c x \| = |(c x)_2| = |c x_2| = |c| |x_2| = |c| \| x \| \).
• 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 \( \| x + y \| \leq \| x \| + \| y \| \).

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 \( \ell^\infty \).

Not Every Metric Comes from a Norm: While every norm induces a metric, the converse is not true.