Inner product spaces

Inner product spaces are mathematical frameworks that extend the concept of angles and lengths to abstract entities like functions or vectors. Think of them as spaces where you can measure "how similar" two elements are through an inner product, which produces a scalar value. This inner product allows you to define notions like length (norm), angles, and orthogonality (perpendicularity), much like in standard geometry. These spaces are essential in fields like physics, engineering, and computer science, providing a rigorous way to analyze and manipulate complex data, signals, or functions within a structured geometric perspective.