Answer to Question #322804 in Functional Analysis for Deepu

Question #322804

For an example of a discontinuous linear functional on a normed linear space


1
Expert's answer
2022-04-04T15:53:09-0400

"The\\,\\,linear\\,\\,space\\,\\,is\\,\\,LC_1\\left[ 0,1 \\right] \\,\\,of\\,\\,continuous\\,\\,functions\\,\\,on\\,\\,\\left[ 0,1 \\right] \\,\\,with\\\\\\left\\| x \\right\\| =\\int_0^1{\\left| x\\left( t \\right) \\right|dt}\\\\The\\,\\,functional\\,\\,is\\\\f\\left( x \\right) =x\\left( 1 \\right) \\\\Then\\\\f\\left( \\alpha x+\\beta y \\right) =\\left( \\alpha x+\\beta y \\right) \\left( 1 \\right) =\\alpha x\\left( 1 \\right) +\\beta y\\left( 1 \\right) =\\alpha f\\left( x \\right) +\\beta f\\left( y \\right) \\\\thus\\,\\,f\\,\\,is\\,\\,linear.\\\\f\\,\\,is\\,\\,not\\,\\,continuous, \\sin ce\\,\\,for\\,\\,\\\\x_n\\left( t \\right) =t^n\\\\\\left\\| x_n \\right\\| =\\int_0^1{\\left| t^n \\right|dt}=\\frac{1}{n+1}\\rightarrow 0,n\\rightarrow \\infty \\\\which\\,\\,means\\,\\,x_n\\rightarrow 0,n\\rightarrow \\infty in\\,\\,LC_1\\left[ 0,1 \\right] \\\\And\\,\\,we\\,\\,have\\\\f\\left( x_n \\right) =1^n=1,f\\left( 0 \\right) =0\\\\thus\\\\f\\left( x_n \\right) \\nrightarrow f\\left( 0 \\right) ,n\\rightarrow \\infty \\\\which\\,\\,means\\,\\,f\\,\\,is\\,\\,discontinuous"


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