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White Noise Integration
Published in Hui-Hsiung Kuo, White Noise Distribution Theory, 2018
Example 13.1. Let c0 consist of all sequences of complex numbers converging to zero. It is a Banach space with the supremum norm. Define a function f: (0, ∞) → c0 by f(u)=(e−u,2e−2u,…,ne−nu,…).
Preliminaries
Published in Hugo D. Junghenn, Principles of Analysis, 2018
(The space of bounded functions). Let X be a nonempty set and let B(X) denote the vector space (under pointwise addition and scalar multiplication) of all bounded functions f:X→K $ f:X\rightarrow \mathbb K $ . The supremum norm or uniform norm on B(X) is defined by f∞=sup{|f(x)|:x∈X}. $$ \left\Vert f\right\Vert_\infty = \sup \big \{|f(x)|: x \in X\big \}. $$
Applications of Metric Fixed Point Theory
Published in Dhananjay Gopal, Poom Kumam, Mujahid Abbas, Background and Recent Developments of Metric Fixed Point Theory, 2017
and it is a complete metric space with the metric induced by the supremum norm defined on X given by: d(x,y)=supt∈I|x(t)-y(t)|,x,y∈C(I,R). $$ d(x,y)=\sup _{t\in I}|x(t)-y(t)|, \ x,y\in C(I,\mathbb R ). $$
Optimal control for semilinear integrodifferential evolution equations in Banach spaces
Published in International Journal of Control, 2023
Mamadou Abdoul Diop, Khalil Ezzinbi, Paul dit Akouni Guindo
We denote by a reflexive Banach space and by a Polish space, which is a separable complete topological space. the Banach space of continuous function from J to with the usual supremum norm , the usual strongly measurable space for . We suppose that is an infinitesimal generator of semigroup . Hence, endowed with the graph norm is the Banach space, which will be denoted by .
Speed of wave propagation for a nonlocal reaction–diffusion equation
Published in Applicable Analysis, 2020
Let be the Banach space of all bounded and uniformly continuous function from to with conventional supremum norm . Set Then is a closed cone in , and is a Banach space under the partial ordering induced by . It is well known that the differential operator generates an analytic semigroup on . The standard parabolic maximum principle (see Corollary 7.2.3 of [12]) implies that the semigroup is strongly positive in the sense that for all t>0 where Int() is the interior of .
Stability of traveling wave fronts for delayed Belousov–Zhabotinskii models with spatial diffusion
Published in Applicable Analysis, 2020
Yanling Meng, Weiguo Zhang, Zhixian Yu
Let be the Banach space of continuous function from into X with the supremum norm and let then is a closed (positive) cone of . As usual, we identify an element as a function from into defined by . For any continuous function We define as follows: Thus, is a continuous function from to . Define and where ; it is easy to know that is globally Lipschitz continuous.