A normed space is a mathematical structure that consists of a vector space equipped with a function called a norm. This norm assigns a non-negative length or size to each vector in the space, allowing for the measurement of distances between vectors. The norm must satisfy specific properties, such as being zero only for the zero vector, being scalable, and obeying the triangle inequality. In a normed space, the concept of convergence and continuity can be defined, making it a fundamental setting in functional analysis. Common examples of normed spaces include Euclidean spaces and function spaces, which are essential in various fields of mathematics and applied sciences.