ThelinearspaceisLC1[0,1]ofcontinuousfunctionson[0,1]with∥x∥=∫01∣x(t)∣dtThefunctionalisf(x)=x(1)Thenf(αx+βy)=(αx+βy)(1)=αx(1)+βy(1)=αf(x)+βf(y)thusfislinear.fisnotcontinuous,sinceforxn(t)=tn∥xn∥=∫01∣tn∣dt=n+11→0,n→∞whichmeansxn→0,n→∞inLC1[0,1]Andwehavef(xn)=1n=1,f(0)=0thusf(xn)↛f(0),n→∞whichmeansfisdiscontinuous
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