## 数学代写|实分析作业代写Real analysis代考|MATH7400

2022年12月26日

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## 数学代写|实分析作业代写Real analysis代考|Infinite Series in Normed Spaces

We define infinite series in a normed space as follows.
Definition 1.2.6 (Convergent Series). Let $\left{x_n\right}_{n \in \mathbb{N}}$ be a sequence of vectors in a normed space $X$. We say that the series $\sum_{n=1}^{\infty} x_n$ converges and equals $x \in X$ if the partial sums $s_N=\sum_{n=1}^N x_n$ converge to $x$, i.e., if
$$\lim {N \rightarrow \infty}\left|x-s_N\right|=\lim {N \rightarrow \infty}\left|x-\sum_{n=1}^N x_n\right|=0 .$$
In this case, we write $x=\sum_{n=1}^{\infty} x_n$, and we also use the shorthands $x=\sum x_n$ or $x=\sum_n x_n$.

In order for an infinite series to converge in $X$, the norm of the difference between $x$ and the partial sum $s_N$ must converge to zero. If we wish to emphasize which norm we are referring to, we may write that $x=\sum x_n$ converges with respect to $|\cdot|$, or we may say that $x=\sum x_n$ converges in $X$.
If $\left{x_n\right}_{n \in \mathbb{N}}$ is a sequence of vectors in $X$, then $\left{\left|x_n\right|\right}_{n \in \mathbb{N}}$ is a sequence of real scalars. What connection, if any, is there between the convergence of the series $\sum x_n$ in $X$ (which is a series of vectors) and convergence of the series $\sum\left|x_n\right|$ (which is a series of scalars)? In order to address this, we introduce the following terminology.

Definition 1.2.7. Let $\left{x_n\right}_{n \in \mathbb{N}}$ be a sequence in a normed space $X$. We say that the series $\sum_{n=1}^n x_n$ is absolutely convergent if $\sum_{n=1}^n\left|x_n\right|<\infty$.

A convergent series need not converge absolutely. For example, consider $X=\mathbb{R}$ and $x_n=(-1)^n / n$. The alternating harmonic series $\sum_{n=1}^{\infty}(-1)^n / n$ converges, but the harmonic series $\sum_{n=1}^{\infty} 1 / n$ does not.

Also, a series that converges absolutely need not converge. One example in the incomplete space $X=C_c(\mathbb{R})$ is constructed in Problem 1.3.11. The next theorem states that if $X$ is complete then every absolutely convergent series in $X$ must converge. Moreover, the converse also holds: In any incomplete normed space there exists a series that converges absolutely yet does not converge, i.e., there exist vectors $x_n \in X$ such that $\sum\left|x_n\right|<\infty$ but $\sum x_n$ does not converge.

## 数学代写|实分析作业代写Real analysis代考|Equivalent Norms

A vector space $X$ can have many different norms. Some of these norms may be “comparable” in the following sense.

Definition 1.2.9 (Equivalent Norms). We say that two norms $|\cdot|_a$ and $|\cdot|_b$ on a vector space $X$ are are equivalent if there exist constants $C_1, C_2>0$ such that
$$C_1|x|_a \leq|x|_b \leq C_2|x|_a, \quad \text { for all } x \in X .$$
The reader should show that if two norms are equivalent, then they determine the same convergence criterion, i.e.,
$$\lim {n \rightarrow \infty}\left|x-x_n\right|_a=0 \Longleftrightarrow \lim {n \rightarrow \infty}\left|x-x_n\right|_b=0 .$$
Conversely, if equation (1.2) holds, then $|\cdot|_a$ and $|\cdot|_b$ are equivalent (for one proof of this, see [Heil18, Thm. 3.6.2]).

We have the following important fact for finite-dimensional spaces (see [Heil18, Thm. 3.7.2]).

Theorem 1.2.10. If $X$ is a finite-dimensional vector space, then any two norms on $X$ are equivalent. $\diamond$

One consequence of Theorem 1.2.10 is that all finite-dimensional subspaces of a normed space are closed (see [Heil18, Cor. 3.7.3]).

# 实分析代写

## 数学代写|实分析作业代写Real analysis代考|Infinite Series in Normed Spaces

Veft $\left{x_{-} n \backslash r i g h t\right}_{-}{n \backslash i n \backslash m a t h b b{N}}$ 是赋范空间中的向量

$$\lim N \rightarrow \infty\left|x-s_N\right|=\lim N \rightarrow \infty\left|x-\sum_{n=1}^N x_n\right|$$

$X$ ，然自 $\backslash$ left $\left{\backslash \text { left } \mid x_{-} n \backslash \text { right } \mid \backslash \text { right }\right}_{-}{n \backslash$ in $\backslash m a t h b b{N}}$

$\sum\left|x_n\right|$ (这是一系列标量) ? 为了解决这个问题，我们 引入了以下术语。

## 数学代写|实分析作业代写Real analysis代考|Equivalent Norms

$$C_1|x|_a \leq|x|_b \leq C_2|x|_a, \quad \text { for all } x \in X$$

$$\lim n \rightarrow \infty\left|x-x_n\right|_a=0 \Longleftrightarrow \lim n \rightarrow \infty\left|x-x_n\right|_b$$

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