Bounded Linear Functionals

A bounded linear functional is a special type of function that takes a vector (a point in a space) and returns a number, while respecting two key properties: linearity (the output for a sum of vectors equals the sum of their outputs, and scaling a vector scales the output by the same factor) and boundedness (there's a limit to how large the output can be relative to the input's size). Essentially, it’s a consistent, well-behaved way of extracting a single numerical measurement from vectors in a space, with control over how much this measurement can change with different inputs.