} This norm is also called the supremum norm, the Chebyshev norm, the infinity norm, or, when the supremum is in fact the maximum, the max norm. The name...

Haar measure). These norms are also valid in the limit as p → + ∞ {\displaystyle p\rightarrow +\infty } , giving a supremum norm, and are called ℓ ∞ {\displaystyle...

denotes the supremum. This norm measures how much the mapping induced by A {\displaystyle A} can stretch vectors. Depending on the vector norms ‖ ⋅ ‖ α {\displaystyle...

exists. If the supremum of S {\displaystyle S} exists, it is unique, and if b is an upper bound of S {\displaystyle S} , then the supremum of S {\displaystyle...

{\displaystyle p>1.} One also defines the ∞ {\displaystyle \infty } -norm using the supremum: ‖ x ‖ ∞ = sup ( | x 1 | , | x 2 | , … , | x n | , | x n + 1 |...

}\|_{\infty })} denote the Banach space of continuous functions with the supremum norm and let τ ∞ {\displaystyle \tau _{\infty }} denote the topology on C...

complete metric space, with respect to the supremum norm. However, the supremum norm does not give a norm on the space C(a, b) of continuous functions...

, the (real or complex) vector space of bounded sequences with the supremum norm, and L ∞ = L ∞ ( X , Σ , μ ) {\displaystyle L^{\infty }=L^{\infty }(X...

functions defined on some set (with pointwise multiplication and the supremum norm) is a unital Banach algebra. The algebra of all bounded continuous real-...

{\displaystyle \ell ^{\infty }} of all bounded real sequences, with the supremum norm, is not separable. The same holds for L ∞ {\displaystyle L^{\infty }}...

L^{\infty }} consisting of bounded functions on the real line with the supremum norm, then the map φ {\displaystyle \varphi } is not surjective. (See L p...

operator norm measures the "size" of certain linear operators by assigning each a real number called its operator norm. Formally, it is a norm defined...

0 {\displaystyle c_{0}} the space of convergent sequences with the supremum norm has the AK property. ℓ p {\displaystyle \ell ^{p}} ( 1 ≤ p < ∞ {\displaystyle...

for all x in [a, b], we have |f(x) − p(x)| < ε, or equivalently, the supremum norm ‖f − p‖ < ε. A constructive proof of this theorem using Bernstein polynomials...

{\displaystyle \ell ^{\infty }} with respect to the supremum norm, and so it is a Banach space with respect to this norm. A sequence that converges to 0 {\displaystyle...

{\displaystyle \Vert \;\Vert _{L^{\infty }(\Omega )}} is the essential supremum norm, and div {\displaystyle \operatorname {div} } is the divergence operator...

linear maps are called "functionals". The norm of a functional φ {\displaystyle \varphi } is defined as the supremum of | φ ( v ) | {\displaystyle |\varphi...

the Banach spaces c (consisting of all convergent sequences, with the supremum norm) and c0 (the sequences converging to zero) are both naturally identified...

Mathematically, the Chebyshev distance is a metric induced by the supremum norm or uniform norm. It is an example of an injective metric. In two dimensions...

equations with local monotonicity and coercivity assumption with respect to supremum norm. Let X ( t ) , t ≥ 0 {\displaystyle X(t),\,t\geq 0} be a non-negative...

{\text{where}}\ \ y\in {\mathcal {C}}(a,b)} is called the uniform norm or supremum norm ('sup norm'). Kolmogorov, A. N., & Fomin, S. V. (1967). Elements of the...

of weighted Hölder norms; the notation will follow that given in the text of D. Gilbarg and Neil Trudinger (1983). The supremum norm of a continuous function...

square is given by the p → ∞ case (the supremum norm), and a rotated square is given by p = 1 (the taxicab norm). This allows a straightforward generalization...

of bounded sequences ℓ ∞ {\displaystyle \ell ^{\infty }} under the supremum norm ‖ ⋅ ‖ ∞ . {\displaystyle \|\cdot \|_{\infty }.} The space of absolutely...

\lim _{p\rightarrow +\infty }W_{p}(\mu ,\nu )} and corresponds to a supremum norm. Here, a coupling γ {\displaystyle \gamma } is a joint probability measure...

φn : [0, T] → X converging uniformly to f (i.e. with respect to the supremum norm || - ||∞). It requires a little work to show that these two conditions...

on the interval [ a , b ] , {\displaystyle [a,b],} equipped with the supremum norm. Every metric space is dense in its completion. Every topological space...

R {\textstyle [0,1]\to \mathbb {R} } . It is a function space with supremum norm (it is a Banach space). Let f {\textstyle f} be a continuous function...

{\displaystyle A_{\infty }(\Omega )} of all bounded analytic functions with the supremum norm is a Banach space. A polynomial cannot be zero at too many points unless...

{\displaystyle f} "best", i.e. with minimum distance to f {\displaystyle f} in supremum norm. Shapiro, Harold (1971). "2. Best uniform approximation". Topics in...