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 specific properties, such as being non-negative and obeying the triangle inequality. These spaces are essential in various fields, including functional analysis and optimization. Common examples of normed vector spaces include Euclidean spaces, where the norm is the standard distance formula, and Banach spaces, which are complete normed vector spaces that play a crucial role in analysis.