- #1
foxjwill
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Homework Statement
Is the derivative operator [tex]D:L^2(0,1)\to L^2(0,1)[/tex] bounded? In other words, is there a c>0 such that for all [tex]f\in L^2(0,1)[/tex],
[tex]\|Df\|\leq c\|f\|?[/tex]
Homework Equations
For all [tex]f\in L^2(0,1)[/tex],
[tex]\|f\| = \int_0^1 |f|^2\,dx.[/tex]
The Attempt at a Solution
I'm pretty sure the answer is no. Here's my work:
Suppose [tex]c^2>0[/tex] satisfies the above requirements. Define [tex]f(x)=e^{(c+1)x}[/tex]. Then
[tex]\|Df\| = \int_0^1 (c+1)^2e^{2(c+1)x}\,dx = (c+1)^2\|f\| > c^2\|f\|.[/tex]
But this contradicts the fact that [tex]\|Df\|\leq c^2\|f\|.[/tex] Thus, D is unbounded. Q.E.D.Is this correct?