Normed Spaces

A normed space is a mathematical setting where vectors (objects representing quantities like forces or directions) exist with a way to measure their size or length, called a "norm." This measurement follows certain rules: it’s always non-negative, only zero if the vector is zero, scales proportionally when the vector is scaled, and satisfies the triangle inequality (the direct distance is always shortest). Normed spaces are fundamental in analyzing concepts like convergence, continuity, and approximation in various areas of mathematics, providing a structured way to quantify and manipulate the size and distance between objects in infinite or finite-dimensional contexts.