} 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...

, 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...

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...

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...

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...

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...

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...

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...

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

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...

{\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...

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

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...

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

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

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

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...

Stein operator), where often ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is the supremum norm. Here, D k {\displaystyle D^{k}} denotes the differential operator,...

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

{\displaystyle f} "best", i.e. with minimum distance to f {\displaystyle f} in supremum norm. Shapiro, Harold (1971). Topics in Approximation Theory. Springer. pp...

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...

concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional...

{\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...

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...

x : N → R with only finitely many non-zero terms equipped with the supremum norm. The map T : X → X defined by T x = ( x 1 , x 2 2 , x 3 3 , … ) {\displaystyle...

sup-norm. If Σ is a sigma-algebra and μ is a sigma-additive positive measure on Σ then the Lp space L∞(μ) endowed with the essential supremum norm is by...

analytic functions u : Ω → C is a Banach space with respect to the supremum norm. Morera's theorem can also be used in conjunction with Fubini's theorem...