Define $T:C([0,1])\rightarrow C([0,1])$ by $T(f)(x)=f(0)+\int_0^xtf(t)dt$ I want to show that $T$ is a continuous linear operator.Showing the linear part is easy enough, but I am not quite sure how to show the continuity part. I am trying to show that there exists $c$ s.t. $||T(f)||\le c||f||$.Or I might need to use that $\{||T(f)(x)|| : \ f(x)\le 1 \}\le \infty$ I am not quite sure... any help would be great.
2026-03-31 04:35:58.1774931758
I need help showing something is a linear continuous operator.
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Clearly $f(0) \le \lVert f\rVert$ so we needn't worry about that piece. The harder part is the integral. The way to do this is to note that:
$$\left|\int_0^x tf(t)\,dt\right|\le\int_0^xt|f(t)|\,dt \le \int_0^x t\lVert f\rVert \,dt.$$
Can you take it from here?