The Supremum Norm

Real Analysis · Axiom Academy

One number measures a whole function's size — and it turns a space of functions into a metric space where convergence IS uniform convergence. For a bounded function f on a set S , the supremum norm is the least upper bound of |f(x)| — the single number no point of the graph exceeds in magnitude. Watch the marker walk the curve: it rides the height |f(x)| , and the bar on the left climbs to the highest reach. That ceiling is . the size of a whole function, in one number 2. Computing It on Real Functions To find , take |f| and hunt for its supremum: check the endpoints and any interior critical point, then keep the largest magnitude. The animation samples each function finely, tags the winning point, and drops the red ceiling there. |x|=x rises to the right, so the max sits at the endpoint x=1 : . Extremes are f(-1)=-1 and f(2)=2 , so . Endpoints critical points take the biggest |f| . No single formula — it's a search. A rule earns the name norm only if it obeys three axioms. The supremum norm passes all three, which is what makes a normed vector space. The animation stacks a real f and g and shows the triangle inequality: the tallest reach of f+g can never beat the two ceilings added. The size is never negative, and it is zero only for the function that is zero everywhere. Scaling a function by c scales its size by |c| — stretch the graph, stretch the reach. At every x , ; take the sup on the left.

This is the written version of the interactive lesson above. See the full Real Analysis course.