Description: Definitions for forms [14 Q], [52 E], [52 N], and 410, 411, and 412.
Content:
Definitions for forms [14 Q], [52 E], [52 N], and 410, 411, and 412.
If \(E\) is a normed linear space or a Banach space (complete,normed, linear space) \(E^{*}\) (the dual of \(E\)) is the space of all bounded (continuous) linear functionals on \(E\). (That is,all \(f : E \rightarrow {\Bbb R}\) such that \((\exists M > 0)(\forall x \in E)(|f(x)| \le \Vert M \Vert\)).) \(E^{*}\) is a Banach space if we define \(\Vert f\Vert=\sup_{x\neq 0}\left(\frac{|f(x)|}{\Vert x\Vert} \right)\). Each \(x\in E\) can be thought of as a linear functional on \(E^{*}\) if we define \(x(f) =f(x)\) for all \(f\) in \(E^{*}\). The weak\(^{*}\) topology on \(E^{*}\) is the weakest topology that makes all these linear functionals continuous. (A sequence \(\{f_n\}\) in \(E^{*}\) is said to be weakly\(^{*}\) convergent if \(\lim_{n\to\infty} f_n(x)\) exists for every \(x\in E\). A sequence \(\{x_n\}\) in \(E\) is said to be weakly convergent if \(\lim_{n\to\infty}f(x_n)\) exists for every \(f\in E^{*}\).) We let \(E^{**}\) be the dual of \(E^{*}\).If \(\phi : E\to E^{**}\) such that for each \(x\in E\), \(\phi(x)= f_x\), where for all \(g\in E^{*}\), \(f_x(g)=g(x)\), then \(\phi\) is called the natural embedding of \(E\) into \(E^{**}\). If the natural embedding is onto, the space is called reflexive.
Sets of the form \(w_{x,\epsilon } =\{ f \in E^{*}: |f(x)| < \epsilon\}\) form a basis for the weak\(^{*}\) topology. If \(X\subseteq E\), \(X\) is convex-compact if whenever \(F_i\), for \(i\in I\), are closed convex subsets of \(X\) and\(\{X\cap F_i: i\in I\}\) has the finite intersection property, then \(\bigcap_{i\in I}(X\cap F_i)\ne\emptyset\). Note that the definition of convex-compact may be given in any topological vector space. \(E\) is said to be uniformly convex if for every \(\epsilon>0\) there is a \(\delta>0\) such that for all \(x, y\in E\) with \(\|x\|=\|y\|=1\), \(\|x-y\|\ge\epsilon\) implies \(\frac12\| x+y\|\le 1-\delta\). (A topological vector space (linear topological space) is a vector space (linear space) with a topology in which the operations of addition and scalar multiplication are continuous. An affine subspace, \(A\), of a topological vector space \(E\) is a translation of a subspace of \(E\), \(A =v+S = \{v + w: w\in S\}\), where \(S\) is a subspace of \(E\) and \(v\in E\).)
(The main reason that form [14 Q] implies Form 410 is that if the space \(E\) is reflexive, then there is a mapping of \(E\) onto \(E^{**}\) and the weak topology on \(E\) corresponds to the weak\(^{*}\) on \(E^{**}\).)
Howard-Rubin number: 23
Type: Definitions
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