Answer to Question #308219 in Functional Analysis for mathforfun

Question #308219

prove ||Ax|| < ||A|| ||x||


1
Expert's answer
2022-03-09T14:29:55-0500

Solution


Let us suppose that


"||Ax|| > ||A|| ||x||"


"\\Rightarrow\\frac{||Ax||}{||x||} > ||A||"


"\\Rightarrow|A\\frac{x}{||x||}| > ||A||"


Since "\\frac{x}{||x||}|" is a unit vector, therefore, from above we can write,


"\\Rightarrow ||A|| > ||A||" which constradicts


Hence


"||Ax|| < ||A|| ||x||"


Proved






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