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Let $X,Y$ be Hilbertspaces, and let $A\colon X\to Y$ be a compact operator. The Tikhonov functional is given by $$ F(x)=\lVert Ax-y\rVert_X^2+\alpha\lVert x\rVert_X^2. $$ Calculate the Gâteaux derivate of the Tikhonov functional.

Tip: Use $\lVert a+b\rVert=\lVert a\rVert^2+2(a,b)+\lVert b\rVert^2$.

First question: Is it really $\lVert Ax-y\rVert_X^2$ and not the Y-norm?

Second question: What I have to do here is to my opinion calculate

$$ \lim\limits_{t\to 0}\frac{F(x+th)-F(x)}{t}. $$

So I would start with calculating

$F(x+th)=\lVert A(x+th)-y)\rVert_X^2+\alpha\lVert x+th\rVert_X^2$.

I start with calculating $\lVert A(x+th)-y)_X^2$. I can not read from the text if the operator $A$ is linear, too. Is it right to calculate with the given tip

$\lVert A(x+th)-y\rVert_X^2=\lVert A(th)+Ax-y\rVert_X^2=\lVert A(th)\rVert_X^2+2(A(th),Ax-y)+\lVert Ax-y\rVert_X^2$?

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The first norm should be in $Y$, not in $X$. In your definition of the Tikhonov functional you may also want to specify what $y$ is. I assume it is a constant vector in $Y$. –  Willie Wong Feb 1 '13 at 13:54
And yes, the notations and terminology suggest that $A$ is linear. –  Willie Wong Feb 1 '13 at 13:55
My result then is: The Gâteaux derivate in $x$ in direction $h$ is given by $DF(x)[h]=2\cdot(h,(A^*A+\alpha\mbox{id})x-Ay)_X$. –  math12 Feb 1 '13 at 14:34

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