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Let $X=C_c(\mathbb{R})$, the space of functions with compact support, normed with the max norm. Define $\Gamma: X \rightarrow \mathbb{R}$ as: $$ \Gamma f= \int_{- \infty}^{\infty} f(t)dt \>\>\> \forall f \in X$$ Prove $\Gamma$ is not continuous.

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Find a sequence of functions $(f_n)$ which are bounded w.r.t. the max norm (for example, their maximum value is $1$), but where $\Gamma f_n$ increases. If you don't need continuity of the functions, then something like the characteristic function of $[-n,n]$ would suffice. –  Roland Jun 3 at 10:14
    
I understood your solution. I was thinking about showing that the kernel of the operator is dense in the space, but I wasn't able to do it...this should stand anyway right? –  user73793 Jun 3 at 10:21
    
@user73793: Why do you think that the kernel is dense? –  John Jun 3 at 10:25

1 Answer 1

Let $f$ be a continuous function such that $f(x)\in [0,1]$ for all $x$, $f(t)=1$ if $t\in [-1,1]$ and $f(t)=0$ if $|t|>2$. Then define $f_n(x):=f(nx)/n$. Since $ \int f_n(x)\mathrm dx=\int_\mathbb R f(x)\mathrm d x\gt 0$ and $\lVert f_n\rVert=1/n$, $\Gamma$ is not continuous.

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