A normed vector space is a mathematical structure that combines the concepts of vectors and norms. In this space, vectors can be added together and multiplied by scalars, while the norm provides a way to measure the length or size of these vectors. The norm must satisfy certain properties, such as being non-negative and obeying the triangle inequality. In a normed vector space, the norm allows for the comparison of vectors, enabling the analysis of convergence and continuity. Common examples of normed vector spaces include Euclidean spaces and function spaces, which are essential in various fields like physics and engineering.