This norm is also called the supremum norm or the Chebyshev norm. If f is a continuous function on a closed interval, or more generally a compact set, then the supremum in the above definition is attained by the Weierstrass extreme value theorem, so we can replace the supremum by the maximum. In this case, the norm is also called the maximum norm.
The occasion for the subscript "∞" is that
where
where D is the domain of f.
The binary function
is then a metric on the space of all bounded functions on a particular domain. A sequence { fn : n = 1, 2, 3, ... } converges uniformly to a function f if and only if
For complex continuous functions over a compact space, this turns it into a C* algebra.
In analysis the supremum or least upper bound of a set S of real numbers is denoted by sup(S) and is defined to be the smallest real number that is greater than or equal to every number in S.
In particular, note the third example where the supremum of a set of rationals is irrational (which means that the rationals are incomplete).
The difference between the supremum of a set and the greatest element of a set may not be immediately obvious.
The algebra of bounded real- or complex-valued functions defined on some set (with pointwise multiplication and the supremumnorm) is a Banach algebra.
The algebra of continuous real- or complex-valued functions on some compact space (again with pointwise operations and supremumnorm) is a Banach algebra.
The algebra of all continuous linear operators on a Banach space (with functional composition as multiplication and the operator norm as norm) is a Banach algeba.
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