A normed space is a type of mathematical structure that helps us understand the concept of distance and size in a more abstract way. It consists of a set of elements, often called vectors, along with a function known as a norm. This norm assigns a non-negative length or size to each vector, allowing us to measure how far apart two vectors are in the space. In a normed space, the norm must satisfy certain properties, such as being zero only for the zero vector and obeying the triangle inequality. Normed spaces are essential in various fields, including functional analysis and linear algebra, as they provide a framework for studying convergence, continuity, and other important concepts in mathematics.