ozkan12 said:Dear Ackbach,
I know this...But what is the nonlinearity ? I have troubles related to this term...?
ozkan12 said:What is the relation between metric spaces and normed spaces... What is the meaning of " metric spaces are seen as a nonlinear version of vector spaces endowed with a norm" ? Thank you for your attention...Best wishes...:)
A metric space is a mathematical structure that consists of a set of objects and a function called a metric that measures the distance between any two objects in the set. This distance function follows certain properties, such as being non-negative, symmetric, and satisfying the triangle inequality.
A normed space is a vector space that has a norm, which is a function that assigns a length or size to each vector in the space. This norm function also follows certain properties, such as being non-negative, scalar multiplication preserving, and satisfying the triangle inequality.
The main difference between a metric space and a normed space is the type of objects they contain. A metric space contains any type of objects, while a normed space contains vectors. Additionally, the distance function in a metric space is between any two objects, while the norm function in a normed space is between a vector and the origin.
Metric spaces and normed spaces are widely used in various scientific fields, such as physics, engineering, computer science, and statistics. They provide a mathematical framework for studying the properties and behavior of objects or vectors in a given space, which is crucial for understanding and solving many scientific problems.
Some common examples of metric spaces include Euclidean space, which is the set of all points in a plane or a three-dimensional space, and graph metric space, which measures the distance between vertices in a graph. Examples of normed spaces include Euclidean space with the standard Euclidean norm and function spaces with the Lp-norm, which measures the size of a function.