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有界算子.
在泛函分析此一數學分支裡,有界線性算子是指在賦範向量空間X 及Y 之間的一種線性變換L,使得對所有X 內的非零向量v,L(v) 的範數與v 的範數間的比值會侷限在相同的數字內。亦即,存在一些M > 0,使得對所有在X 內的v,
其中最小的M 稱為L 的算子范数。。
有界線性算子一般不會是有界函數;後者需要對所有的v,L(v)的範數是有界的,但這只有在Y 為零向量空間時才有可能。然而,有界線性算符為局部有界函數。
一個線性算子為有界的,若且唯若其為連續的。因此有界线性算子也被称为连续线性算子。
- 任何在兩個有限維度賦範空間之間線性算符皆是有界的,且此類算符可以被視為某些固定矩陣的乘積。
- 許多積分變換為有界線性算符。例如,設
- 為一連續函數,則算符L
- (定義於由在 上的連續函數所組成的空間,賦予空間 均勻範數的值)是有界的。此一算符實際上也是緊緻的。緊緻算符在有界算符中是很重要的一類。
- (其定義域為索伯列夫空間,值域在由平方可積函數所組成的空間內)是有界的。
- 在由所有實數序列(x0, x1, x2...)(其中)所組成的l2 空間上的位移算符
- 是有界的。其算符範數可輕易地看出為1。
如開頭所述,在賦範空間X 及Y間的線性算子L 是有界的,若且唯若其為連續線性算子。證明如下:
- 設L 是有界的,則對X內的所有向量v 及h(其中的h不為零),會有
- 。
- 令 趨近於零,即可證明L 在v 是連續的。甚至,因為常數M 不依賴v,可證明L 實際上是均勻連續的(更甚之,還是利普希茨連續的)。
- 反過來,在零向量的連續性,允許存在一個,使得對所有X 內 的向量h,。因此,對所有'X 內的非零向量v,會有
- 這證明了L 是有界的。
- Kreyszig, Erwin: Introductory Functional Analysis with Applications, Wiley, 1989
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