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Linear Metric Spaces
Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019
Eberhard Malkowsky, Vladimir Rakočević
Chapter 3 deals with linear metric and linear semimetric spaces, the concepts of paranormed spaces and Schauder bases, and their most important properties. Among other things, it is shown that the quotient of a paranormed space is a paranormed space, and that quotient paranorms preserve completeness. The highlights of the chapter are the open mapping theorem, the closed graph theorem, the uniform boundedness principle and the Banach–Steinhaus theorem, which are generally considered as being the main results in functional analysis, apart from the Hahn–Banach extension theorem which is presented in Chapter 4. Furthermore, the chapter contains studies of useful properties of seminorms, further results related to local convexity, and the Minkowski functional and its role in defining a seminorm or norm on a linear space. Finally, a sufficient condition is established for the metrizability of a linear topology, and also a criterion is given for a linear topology to be generated by a seminorm.
Banach Spaces
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
Seminorm. Let V be a vector space. Recall that a function p:V→[0,∞) $ p :V \rightarrow [0,\infty ) $ is called a seminorm iffp(αu)=|α|p(u) $ p (\alpha \boldsymbol{u}) = |\alpha | p (\boldsymbol{u}) $ (homogeneity)p(u+v)≤p(u)+p(v) $ p (\boldsymbol{u}+ \boldsymbol{v}) \le p (\boldsymbol{u}) + p (\boldsymbol{v}) $ (triangle inequality)for every scalar α $ \alpha $ and vectors u,v $ \boldsymbol{u},\boldsymbol{v} $ . Obviously every norm is a seminorm but not conversely.
Segmented pseudometrics and four-point Fermat-Torricelli problems
Published in Optimization, 2023
Frank Plastria, Francisco Guevara
A seminorm on V is a non-negative, real-valued function that is positively homogeneous ( for all ) and sub-additive (); it is a norm if in addition it is definite (). For example, for any basis , subset and we have the associated -seminorm: which is a norm on V as soon as A = B, in which case we simply write . For p = 2 and A = B we obtain the classical Euclidean norm and we call the norm ellipsoidal because its unit ball is an ellipsoid, or, equivalently, the norm is derived from a scalar product.
A fourth-order difference scheme for the fractional nonlinear Schrödinger equation with wave operator
Published in Applicable Analysis, 2022
Kejia Pan, Jiali Zeng, Dongdong He, Saiyan Zhang
The solution domain is defined as , which is covered by a uniform grid with spacing . For any grid function , the following notations are introduced Denote For any grid functions , we define the discrete inner product and the associated -norm The discrete -norm is defined as Let , and for any given , the fractional Sobolev norm and seminorm are defined as where the relation between the semi-discrete Fourier transform and the grid function are given by Obviously, . Let , then we introduce the following two lemmas which are shown in [25,26,31,32].
A linearized Crank–Nicolson Galerkin FEMs for the nonlinear fractional Ginzburg–Landau equation
Published in Applicable Analysis, 2019
Zongbiao Zhang, Meng Li, Zhongchi Wang
In this example, we set the interval . The initial value u(x, 0) is taken by (5.1) for . As we all know, for , the exact solution cannot be given explicitly, and thus we give the numerical exact solution by using the proposed scheme with a very fine mesh size and time step . Let be the numerical solution. The corresponding convergence orders of the error at time with the norm and the seminorm are calculated by