# 数学代写|数学分析代写Mathematical Analysis代考|Weak Topologies

#### Doug I. Jones

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## 数学代写|数学分析代写Mathematical Analysis代考|Weak Topologies

The weak topologies are defined in much the same way the product topology is defined. They are designed to guarantee the continuity of a certain class of functions. We urge the reader to look up theorem 5.4.1, the definition of the product topology in section 5.12, and theorem 5.12.1. This section is terminal and may be omitted without loss of continuity.

Definition. Let $X$ be a normed linear space. The weak topology on $X$ is the smallest topology relative to which all the bounded linear functionals on $X$ are continuous. We use the abbreviation $w$-topology for the weak topology on $X$.
Definition. Let $X$ be a normed linear space, and let $X^$ be its dual. The weak topology on $X^$ is the smallest topology on $X^$ relative to which the functionals $\hat{x}$ are continuous. Here $\hat{x}$ is the image of $x \in X$ under the natural embedding of $X$ into $X^{ }$. We use the abbreviation $w^$-topology for the weak topology on $X^$. Notice that the definitions of the $w$-and $w^$-topologies are asymmetric. Only the functional on $X^$ of the form $\hat{x}$ is admitted in the definition of the $w^$-topology on $X^$. Thus if $X$ is not reflexive, then the functionals in $X^{ }-\hat{X}$ are not guaranteed to be continuous in the $w^$-topology, and indeed they are not. See theorem 6.7.6.
In order to eliminate any potential confusion, we specifically refer to the topology generated by the norm on a space $X$ (or its dual $X^$ ) as the norm topology on $X$ (or $X^$ ). The norm topology is also referred to as the strong topology. We denote the closed unit balls of a normed linear space $X$ and its dual $X^$ by $B$ and $B^$, respectively. We use notation such as $\left(B^, w^\right)$ to indicate the closed unit ball of $X^$, when it is endowed with the $w^$-topology.

## 数学代写|数学分析代写Mathematical Analysis代考|Definitions and Basic Properties

Let $\left{u_1, u_2, \ldots\right}$ be an infinite orthonormal sequence of vectors in an inner product space $H$, and let $x \in H$. In the introduction to section 4.10 , we posed the following problem. Under what conditions does the sequence of orthogonal projections, $S_n x=\sum_{i=1}^n\left\langle x, u_i\right\rangle u_i=\sum_{i=1}^n \hat{x}_i u_i$, of $x$ on the finite-dimensional space $M_n=\operatorname{Span}\left(\left{u_1, \ldots, u_n\right}\right)$, converge to $x$. Regardless of whether $S_n x$ converges to $x$, it is a Cauchy sequence. To see this, recall the result of problem 5 on section 3.7 (also see theorem 7.2.6,) which states that $\sum_{n=1}^{\infty}\left|\hat{x}n\right|^2<\infty$. Now, for $m>n$, $\left|S_m x-S_n x\right|^2 \leq \sum{i=n+1}^m\left|\hat{x}i\right|^2$. The sum in the last expression tends to 0 as $n \rightarrow \infty$ because it is the middle section of the convergent series $\sum{i=1}^{\infty}\left|\hat{x}_i\right|^2$. Thus we have a sufficient condition for the convergence of the sequence $S_n x$ : the completeness of $H$. This is exactly the definition of a Hilbert space. The completeness of $H$ merely guarantees the convergence of $S_n x$. It does not guarantee that $\lim _n S_n x=x$, as the following situation illustrates. If $u \in H$ is unit vector orthogonal to each $u_n$, then $S_n u=0$ for all $n \in \mathbb{N}$; hence $\lim _n S_n u=0 \neq u$. To remedy this situation, one may want to impose the condition that no such vector $u$ exists. Equivalently, this means that the sequence $\left{u_1, u_2, \ldots\right}$ is a maximal orthonormal subset of $H$, and this is precisely the definition of a countable orthonormal basis for $H$. Hilbert spaces and orthonormal bases are the subject of our study in this section and the next. The question about the smallest Hilbert space $H$ in which trigonometric series of functions in $H$ converge will be settled in section 8.9 , together with related questions pertaining orthogonal polynomials. It is strongly recommended that you study sections 3.7 and 4.10 before you tackle this chapter.

# 数学分析代考

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