A normed vector space is a vector space equipped with a function called a norm, which assigns a non-negative length or size to each vector, except for the zero vector which is assigned a norm of zero. This structure allows for the generalization of various analytical concepts like distance and convergence, making it fundamental in functional analysis and applicable in many mathematical and engineering disciplines.

Normed Vector Space

normed vector space

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

function called a norm

non-negative length

size to each vector

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

norm of zero

generalization of various analytical concepts

distance and convergence

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.

CUSTOMIZE YOUR LEARNING

CUSTOMIZE YOUR LEARNING

Relevant Fields: