02 - 4 SPRING 2012 2 Linear Maps Operator Norm It's not...

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4 SPRING 2012 2. Linear Maps, Operator Norm It’s not obvious from the deFnition that ° T ° is Fnite for all T L ( R n , R m ). If [ T ]=( a ij ) is the matrix representation of T with respect to the standard bases, deFne the 2 -norm of T to be ° T ° 2 = ° ± ± ² m ³ i =1 n ³ j =1 a 2 ij . Then for each x R n ,wehave ° T ( x ) ° 2 = ° [ T ] x ° 2 = m ³ i =1 ´ n ³ j =1 a ij x j µ 2 m ³ i =1 ´ n ³ j =1 a 2 ij n ³ k =1 x 2 k µ = ´ m ³ i =1 n ³ j =1 a 2 ij µ n ³ k =1 x 2 k = ° T ° 2 2 ° x ° 2 . (The inequality follows from the Cauchy-Schwarz inequality.) Thus ° T ° 2 is an upper bound for the set T ( x ) °|° x ° =1 } , so ° T °≤° T ° 2 , and in particular ° T ° is Fnite. Example 2.1. ±or the identity map Id: R n R n ,wehave ° Id ° 2 = ··· = n. However, for any x R n with ° x ° =1 ,wehave ° Id( x ) ° = ° x ° =1 , and it follows that ° Id ° = 1. Exercise. Prove that every T L ( R n , R m ) is uniformly continuous. Lemma 2.2. For each S L ( R m , R ° ) and T L ( R n , R m ) , ° S T °≤° S °° T °

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02 - 4 SPRING 2012 2 Linear Maps Operator Norm It's not...

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