Norms of Functions

Definition

The norm of a function generalizes the concept of length from geometric vectors to the space of functions.

Why It Matters

Defining the “length” of a function is crucial in functional analysis, quantum mechanics, and signal processing. It gives us a precise way to measure error and the “size” of a signal.

Core Concepts

• Norm (L2): $\|f\| = \sqrt{\int_a^b |f(x)|^2 dx}$.
  • How to read: “The norm of the function f is equal to the square root of the integral of the function f of x squared with respect to x.”
  • Meaning: L2 norm is the “length” of a function—root mean square over the interval.
• Distance Between Functions: The norm of the difference, $\|f - g\|$, provides a rigorous measure of distance or error between two functions.

Connected Concepts

• Inner Products of Functions
• Function Definition, Function Notation
• Function Domain
• Function Range