英语翻译 牛人进3.2 Domain , Range , Kernel ,and the Inverse Operato

英语翻译 牛人进
3.2 Domain , Range , Kernel ,and the Inverse Operator
It may be the case that an operator is not defined on the whole of X . For example , in Exercise 3.2 you are asked to show that the integration operator
公式5 is bounded from C ([0,1]) into L (0,1), where both spaces are equipped with the L (0,1) norm (i.e. we consider C ([0,1]) as a subspace of L (0,1)). In other words we define I[f] for functions that are continuous, but we consider questions of continuity by using the norm in L . We call this subspace on which the operator is defined its domain , and for a general linear operator A we write it as D(A).
If A : X Y, then the image of the domain of A under application of A is called the range of A ,written R(A):
一条公式
This may well be a proper subspace of Y , as in the example of the integration operator , where the range is
一条公式
In general we could write ,perhaps uninstructively,
一条公式
A bounded linear operator defined on a linear subspace of X can be extended to be a bounded linear operator defined on the whole of X . (This is not obvious . We still study a related result , the Hahn-Banach theorem, in the next chapter.) Because of this it is somewhat artificial to restrict the domain of definition of a bounded linear operator , and in the rest of our discussion we will assume that D(A) =X . However, when we come to discuss unbounded operators in the final sections of this chapter, the domain will form an intrinsic part of the definition of the operator.
As with the theory of matrices, the concept of the inverse of a general linear operator is extremely useful. We say that A is invertible if the equation Ax=y has a unique solution for every y R(A)(i.e. if A is injective). In this case we define the inverse of A, A by A y=x. If A is linear and A exists then it is linear too (see exercise 3.3).
Another important concept is the kernel of A , Ker(A), the space of all elements of D(A) that A sends to zero : Ker(A)={u D(A):Au=0}. The invertibility of A is equivalent to the triviality of its lernel.
Lemma 3.4 A is invertible iff Ker (A) ={0}.
Proof: Suppose that A is invertible ,Then the equation Ax=y has a unique solution for any y R(A). However , if Ker(A) contains some nonzero element z then A(x+z)=y also, so Ker(A) must be {0}. Conversely, if A is not invertible then for some y R(A) there are two distinct solutions , x and x , of Ax=y, and so A(x - x )=0,giving a nonzero element of Ker(A).
This characterization of invertibility will prove useful later.

3.2领域、范围、仁和反算子
It也许是实际情形操作员总体上没有被定义例如X.,在锻炼您请求显示那综合化操作员的3.2
公式5从C一定([0,1])入L (0,1),其中两空间用L (0,1)准则(即我们装备考虑C ([0,1])作为子空间L (0,1)).换句话说我们定义了I [f]通过使用在L.的准则是连续的作用的,但是我们考虑连续性的问题.我们叫操作员被定义它的领域的这个子空间,并且为一个一般线性操作符A我们写它作为D (A).
If A ：X - Y,A领域的图象在A的应用之下然后叫A的范围,书面R (A) ：
一条公式
This说不定是Y一个适当的子空间,在综合化操作员的例子中,范围
一条公式
我们可能一般来说写,或许uninstructively,
一条公式 在X一个线性子空间定义的A有界线性算子可以被延伸是有界线性算子总体上被定义X.(这不是显然的.我们仍然学习一个相关结果,Hahn-Banach定理,在下个章节.) 因此它是有些人为的制约定义域有界线性算子的,并且在其余我们的讨论我们将假设那D (A) =X.然而,当我们在本章的最后的部分来谈论无边际的操作员,领域将构成操作员的定义的内在部分.与矩阵的理论的As,一个一般线性操作符的反面的概念是非常有用的.我们说A是可转位的,如果等式Ax=y有每y的R一种独特的解答(A) (即,如果A单射).在这种情况下我们定义了A,由A y=x.的A反面.如果A是线性的,并且A存在然后它太是线性的(参见锻炼3.3).
Another重要概念是A,Ker (A),A寄发到零D的所有元素的空间仁(A) ：Ker (A)= {u D (A) ：Au=0}.A的invertibility与它的lernel的琐事是等效的.
Lemma 3.4 A是可转位iff Ker (a) = {0}.
Proof ：假设A是可转位的,然后等式Ax=y有所有y的R (A)一种独特的解答.然而,如果Ker (A)包含某一非零元素z然后A (也x+z)=y,因此Ker (A)必须是{0}.相反地,如果A为某一y R (A)不是可转位的然后有二种分明解答,x和x,Ax=y和如此A (x - x)=0,给Ker (A)的一个非零元素.invertibility的This描述特性将证明有用以后.