标签：des os io for ar cti line amp
Metric spaces is a large class of spaces on which the closeness of two points is depicted by a distance function, or called a metric. Metric spaces are used to depict the convergence. In the following text, we will construct metrics which are compatible with the algebraic structure. In some sense, "Functional Analysis" = "Algebraic Structure" + "Metric Structure".
The common algebraic structures are linear space, algebra, ring, etc.
The common metric structures are norm, inner product, etc.
Suppose $X$ is a set, $d:X\times X\rightarrow \mathbb{R}_+$ satisfies
\begin{equation}
\begin{split}
&(1)\quad d(x,y)=d(y,x),\forall x,y\in X,\\
&(2)\quad d(x,y)\geq 0 \text{ and } d(x,y)=0 \text{ if and only if } x=y,\\
&(3)\quad d(x,y)+d(y,z)\geq d(x,z).
\end{split}
\end{equation}
Then $d$ is called a metric on $X$ and $(X,d)$ is called a metric space.
$C[0,1]$ endowed with the metric:
\begin{equation}
d(x,y)\triangleq \max_{0\leq t\leq1}|x(t)-y(t)|
\end{equation}
is a metric space. And the metric $d$ describes the uniform convergence.
Suppose $(X,d)$ is a metric space. Then $(x_n)\subset X$ is called a Cauchy sequence if $\forall \varepsilon>0$, there exists $N$ such that when $m,n>N$,
\begin{equation}
d(x_m,x_n)<\varepsilon.
\end{equation}
If each Cauchy sequence admits a limit in $X$, then $(X,d)$ is called complete.
$\{F_n\}$ is called a series of decreasing closed sets , if $F_n$ is closed, $F_{n+1}\subset F_n$ and $diamF_n\downarrow0$. Here
\begin{equation}
diam F\triangleq\sup_{x,y\in F}d(x,y).
\end{equation}
Suppose $(X,d)$ is a metric space. Then $X$ is complete if and only if every series of decreasing closed sets $\{F_n\}$ admits a unique common element.
If $X$ is complete and $\{F_n\}$ is a series of decreasing closed sets, we can choose $x_n\in F_n$. Then $x_n$ is a Cauchy sequence. So we can assume $x_n\rightarrow x_0$. It is easy to see that
\begin{equation}
\bigcap_{n\geq 1}F_n=\{x_0\}.
\end{equation}
On the other hand, for every Cauchy sequence $(x_n)$, we set
\begin{equation}
r_n=\sup_{m\geq n}d(x_m,x_n).
\end{equation}
Define
$$F_n=\{x\in X:d(x,x_n)\leq r_n\}.$$
Then it is easy to check that $\{F_n\}$ is a series of decreasing closed sets and thus admits a common element $x_0$. Therefore, $x_n\rightarrow x_0$.
标签：des os io for ar cti line amp
原文地址：http://www.cnblogs.com/schrodinger/p/3891389.html
