A vector norm is a function that assigns a strictly positive length or size to each vector in a vector space, except for the zero vector, which is assigned a length of zero. It is a fundamental concept in linear algebra and functional analysis, providing a quantitative measure of vector magnitude and enabling the comparison of vector sizes and directions.

Vector Norm

vector norm

Function that assigns a length

strictly positive length

A vector space is a mathematical structure formed by a collection of vectors, which can be added together and multiplied by scalars, adhering to specific axioms such as associativity, commutativity, and distributivity. It provides the foundational framework for linear algebra, enabling the study of linear transformations, eigenvalues, and eigenvectors, which are crucial in various fields including physics, computer science, and engineering.

vector space

The zero vector is a vector in a vector space that has all of its components equal to zero, serving as the additive identity. It plays a crucial role in linear algebra as it is the only vector that, when added to any other vector, leaves the other vector unchanged.

zero vector

length of zero

fundamental concept in linear algebra

Functional Analysis is a branch of mathematical analysis that studies spaces of functions and their properties, often using the framework of vector spaces and linear operators. It provides the tools and techniques necessary to tackle problems in various areas of mathematics and physics, including differential equations, quantum mechanics, and signal processing.

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