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Let $X$ be the normed space of all real polynomails in one variable, with $||f||=\int_0^1 |f(t)|dt$. Put $B(f,g)=\int_0^1 f(t)g(t)dt$, and show that $B$ is a bilinear functional on $X\times X$ which is separately continuous but is not continuous.

I have done for proving that $B$ is the bilinear functional and seperately continuous. But I cannot show that $B$ is not continuous. Could someone help me, please? Thanks advance.

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up vote 2 down vote accepted

If $f_n(x)=\sqrt{n}x^n$, then $f_n\to 0$ but $B(f_n,f_n)\to\frac{1}{2}$.

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