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# 数学代写|线性代数代写Linear algebra代考|The Countably Infinite Dimensional Case

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## 数学代写|线性代数代写Linear algebra代考|The Countably Infinite Dimensional Case

In the countably infinite case, we will be dealing with infinite sums, and so questions of convergence will arise. Thus, we begin with the following.

Theorem 13.17 Let $\sigma=\left{\mathbf{u}1, \mathbf{u}_2, \ldots\right}$ be a countably infinite orthonormal set in a Hilbert space H. The series $$\sum{k=1}^{\infty} r_k \mathbf{u}{\mathbf{k}}$$ converges in $\mathrm{H}$ if and only if the series $$\sum{k=1}^{\infty}\left|r_k\right|^2$$
converges in $\boldsymbol{R}$. If these series converge, then they converge unconditionally (that is, any series formed by rearranging the order of the terms also converges). Finally, if the series (13.4) converges then
$$\left|\sum_{k=1}^{\infty} r_k \mathbf{u}{\mathbf{k}}\right|^2=\sum{k=1}^{\infty}\left|r_k\right|^2$$
Proof. Denote the partial sums of the first series by $\mathbf{s}{\mathrm{n}}$ and the partial sums of the second series by $\mathbf{p}{\mathbf{n}}$. Then for $\mathrm{m} \leq \mathrm{n}$

$$\left|s_n-s_m\right|^2=\left|\sum_{k=m+1}^n r_k u_k\right|^2=\sum_{k=m+1}^n\left|r_k\right|^2=\left|p_n-\mathbf{p}{\mathrm{m}}\right|$$ Hence $\left(s_n\right)$ is a Cauchy sequence in $H$ if and only if $\left(\mathbf{p}_n\right)$ is a Cauchy sequence in $\mathbb{R}$. Since both $\mathrm{H}$ and $\mathbb{R}$ are complete, $\left(\mathrm{s}{\mathrm{n}}\right)$ converges if and only if $\left(\mathbf{p}_{\mathbf{n}}\right)$ converges.

If the series (13.5) converges, then it converges absolutely, and hence unconditionally. (A real series converges unconditionally if and only if it converges absolutely.) But if (13.5) converges unconditionally, then so does (13.4). The last part of the theorem follows from the continuity of the norm.

## 数学代写|线性代数代写Linear algebra代考|The Arbitrary Case

To discuss the case of an arbitrary orthonormal set $\Theta=$ $\left{\mathbf{u}{\mathbf{k}} \mid \mathrm{k} \in \mathrm{K}\right}$, let us first define and discuss the concept of the sum of an arbitrary number of terms. (This is a bit of a digression, since we could proceed without all of the coming details – but they are interesting.) Definition Let $\mathscr{G}=\left{\mathbf{x}{\mathbf{k}} \mid k \in K\right}$ be an arbitrary family of vectors in an inner product space $V$. The sum $\sum_{k \in K} x_k$ is said to converge to a vector $\mathbf{x} \in \mathbf{V}$, and we write
$$x=\sum_{k \in K} x_k$$
if for any $\epsilon>0$, there exists a finite set $S \subset K$ for which
$$\mathrm{T} \supset \mathrm{S}, \mathrm{T} \text { finite } \Rightarrow\left|\sum_{\mathrm{k} \in \mathrm{T}} \mathbf{x}{\mathbf{k}}-\mathbf{x}\right| \leq \epsilon$$ For those readers familiar with the language of convergence of nets, the set $\mathscr{P}_0(\mathrm{~K})$ of all finite subsets of $\mathrm{K}$ is a directed set under inclusion, and the function $$\mathrm{S} \rightarrow \sum{k \in S} x_k$$
is a net in H. Convergence of (13.7) is convergence of this net. In any case, we will refer to the preceding definition as the net definition of convergence.

It is not hard to verify the following basic properties of net convergence for arbitrary sums.

Theorem 13.19 Let $\mathscr{G}^G=\left{x_k \mid k \in K\right}$ be an arbitrary family of vectors in an inner product space $V$. If
$$\sum_{k \in K} x_k=x \text { and } \sum_{k \in K} y_k=y$$
then

1) $\sum_{k \in K} r x_k=r x$ for any $r \in F$
2) $\sum_{k \in K}\left(x_k+y_k\right)=\mathbf{x}+\mathbf{y}$
3) $\sum_{\mathrm{K}}\left\langle\mathbf{x}{\mathbf{k}}, \mathbf{y}\right\rangle=\langle\mathbf{x}, \mathbf{y}\rangle$ and $\sum{\mathrm{K}}\left\langle\mathbf{y}, \mathbf{x}_{\mathbf{k}}\right\rangle=\langle\mathbf{y}, \mathbf{x}\rangle$
The next result gives a useful description of convergence, which does not require explicit mention of the sum.

## 数学代写|线性代数代写Linear algebra代考|The Countably Infinite Dimensional Case

$$\sum k=1^{\infty} r_k \mathbf{u k}$$

$$\sum k=1^{\infty}\left|r_k\right|^2$$

$$\left|\sum_{k=1}^{\infty} r_k \mathbf{u k}\right|^2=\sum k=1^{\infty}\left|r_k\right|^2$$

$$\left|s_n-s_m\right|^2=\left|\sum_{k=m+1}^n r_k u_k\right|^2=\sum_{k=m+1}^n\left|r_k\right|^2=\left|p_n-\mathbf{p m}\right|$$

## 数学代写|线性代数代写Linear algebra代考|The Arbitrary Case

$\backslash \operatorname{mathscr}{G}=\backslash$ left ${\backslash \mathrm{mathbf}{x}{\backslash \operatorname{mathbf}{k}} \backslash \operatorname{mid} k \backslash$ in $k \backslash \operatorname{right}}$ 是内积空间中的任意向量族 $V$. 总和
$\sum_{k \in K} x_k$ 据说收敛于一个向量 $\mathbf{x} \in \mathbf{V}$ ，我们写
$$x=\sum_{k \in K} x_k$$

$$\mathrm{T} \supset \mathrm{S}, \mathrm{T} \text { finite } \Rightarrow\left|\sum_{\mathrm{k} \in \mathrm{T}} \mathbf{x} \mathbf{k}-\mathbf{x}\right| \leq \epsilon$$

$$\sum_{k \in K} x_k=x \text { and } \sum_{k \in K} y_k=y$$

1) $\sum_{k \in K} r x_k=r x$ 对于任何 $r \in F$
2) $\sum_{k \in K}\left(x_k+y_k\right)=\mathbf{x}+\mathbf{y}$
3) $\sum_{\mathrm{K}}\langle\mathbf{x k}, \mathbf{y}\rangle=\langle\mathbf{x}, \mathbf{y}\rangle$ 和 $\sum \mathrm{K}\left\langle\mathbf{y}, \mathbf{x}_{\mathbf{k}}\right\rangle=\langle\mathbf{y}, \mathbf{x}\rangle$

## MATLAB代写

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