Answer to Question #322804 in Functional Analysis for Deepu

Question

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

Answer 1

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