Can a Function Go to Zero on NORM 1 but Not on NORM 2 on C[a,b]?

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how to think about a function on C[a,b] which goes to zero on NORM 1
but NOT on NORM2

i got a solution but it very complicated

i don't know how to predict how integrral will workout

i need a function and intervals
 
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You might tell us what norms are "Norm 1" and "Norm 2" instead of having us guess.
 
norm one is like iner product integral deffinition
but norm

norm 2 is the squere of the absolute value in that integral
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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