Norm

Definition

A norm is a function that assigns a non-negative length or size to vectors in a vector space, satisfying specific properties such as absolute homogeneity, the triangle inequality, and definiteness. Norms are fundamental in various mathematical contexts, including inner product spaces where they are derived from inner products, and in Banach spaces where they help define complete metrics for convergence and continuity.

5 Must Know Facts For Your Next Test

1. A norm must satisfy three essential properties: it must be zero only for the zero vector, it must be absolutely homogeneous (scaling a vector scales its norm), and it must satisfy the triangle inequality (the norm of the sum is less than or equal to the sum of the norms).
2. In inner product spaces, the norm can be derived from the inner product using the formula $$||x|| = \sqrt{\langle x, x \rangle}$$.
3. Every Banach space has a corresponding norm, which defines its structure and allows us to discuss concepts like convergence, continuity, and boundedness.
4. Hilbert spaces are specific types of inner product spaces that are also complete with respect to their norm, making them essential in functional analysis.
5. The concept of orthogonality in Hilbert spaces is defined using norms, as two vectors are orthogonal if their inner product is zero, leading to useful properties in projections and decompositions.

Review Questions

• How does the definition of a norm relate to inner product spaces and what implications does this have for vector lengths?
  • The definition of a norm is closely linked to inner product spaces because the norm can be derived from the inner product. Specifically, in an inner product space, the norm of a vector is calculated as the square root of its inner product with itself. This relationship allows us to define lengths of vectors in geometric terms and enables further study of angles and orthogonality between vectors.
• Discuss how the concept of norms plays a role in determining whether a space is classified as a Banach space.
  • A Banach space is defined as a complete normed vector space. This means that not only must there be a norm defined on the space, but every Cauchy sequence within that space must converge to an element also within the space. The existence of a suitable norm that satisfies all properties is crucial for ensuring that limits and convergence behaviors align with our expectations in functional analysis.
• Evaluate how understanding norms enhances our ability to work with compact operators in functional analysis.
  • Understanding norms is vital when working with compact operators because these operators behave uniquely in relation to bounded sets within Banach spaces. Norms allow us to measure distances and determine convergence properties associated with sequences and functionals. Since compact operators map bounded sets to relatively compact sets, knowing how norms interact with these mappings can help analyze continuity and stability within different functional frameworks.

Related terms

Inner Product:

An inner product is a generalization of the dot product that defines a geometric structure on a vector space, allowing the measurement of angles and lengths.

Banach Space:

A Banach space is a complete normed vector space where every Cauchy sequence converges within the space, providing a framework for functional analysis.

Compact Operator:

A compact operator is a linear operator on a Banach space that maps bounded sets to relatively compact sets, which is essential for understanding certain convergence properties.