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# 数学代写|数学分析作业代写Mathematical Analysis代考|Three Fundamental Theorems

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## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Three Fundamental Theorems

In addition to the Hahn-Banach theorem, the three theorems we present in this section are of fundamental importance. All three theorems require completeness; hence they apply only to Banach spaces.
In chapter 4 (see problem 5 on section 4.8 ), we encountered an example where a family of pointwise bounded functions on a complete metric space is, in fact, uniformly bounded on a ball. Lemma 6.3.1 is similar in spirit, and its proof demonstrates the centrality of Baire’s theorem in this section. Because the boundedness of a linear function on a ball implies its boundedness, it must not be surprising that when $X$ is a Banach space, pointwise boundedness implies uniform boundedness. This is the uniform boundedness principle.
The open mapping theorem is a central theorem in functional analysis, and one cannot exaggerate its importance. Lemma 6.3 .3 is critical to the proof of the open mapping theorem, and, again, completeness is crucial. The closed graph theorem comes in quite handy in certain applications to prove the boundedness of a linear function. It follows rather easily from the open mapping theorem. Later in the book, you will see many applications of the three theorems, as well as the Hahn-Banach theorem.
In this section, $X$ and $Y$ are normed linear spaces.
A family of bounded linear functions $\left{T_\alpha\right}_{\alpha \in I}$ from $X$ to $Y$ such that, for each $x \in X$, $\sup {\alpha \in I}\left{\left|T\alpha(x)\right|\right}<\infty$ is said to be pointwise bounded. If $\sup {\alpha \in I}\left|T\alpha\right|<\infty$, we say that the family $\left{T_\alpha\right}$ is uniformly bounded.

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|The Hahn-Banach Theorem

The importance of the Hahn-Banach theorem cannot be overstated. The results following theorem 6.4.4 represent only a sample of the wide range of applications of the Hahn-Banach theorem. Unlike the three major theorems of the previous section, the Hahn-Banach theorem does not require completeness.
The Hahn-Banach theorem has many guises, and one of them is an extension theorem. The following example shows that, from the purely algebraic perspective, extending a linear functional on a subspace $M$ of a vector space $X$ is a trivial task. Compare the following example to theorem 6.4.4.
Example 1. Let $M$ be a subspace of a vector $\operatorname{space} X$, and let $\lambda$ be a linear functional on $M$. Then $\lambda$ can be extended to a linear functional on $X$.
Let $S_1$ be a basis for $M$, and choose a subset $S_2$ of $X$ such that $S_1 \cup S_2$ is a basis for $X$. Define a function $\Lambda: S \rightarrow \mathbb{C}$ as follows:
$$\Lambda(x)= \begin{cases}\lambda(x) & \text { if } x \in S_1, \ 0 & \text { if } x \in S_2\end{cases}$$
Extend the function $\Lambda$ by linearity to a functional $\Lambda$ on $X$. The restriction of $\Lambda$ to $M$ is clearly $\lambda$.

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|The Hahn-Banach Theorem

$$\Lambda(x)= \begin{cases}\lambda(x) & \text { if } x \in S_1, \ 0 & \text { if } x \in S_2\end{cases}$$

## MATLAB代写

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