Example of calculation of Frechet derivative considering Hilbert space

Question

Let $\mathbb R$ a Hilbert space in their canonic form. We define the function $f(x)=a\|x-x^{o}\|^{2}+\langle b,x \rangle $, $\forall x \in \mathbb R^{n}$, where $a$ is a constant, while $x^{o}$ and $b$ are vectors of $\mathbb R^{n}$.

Find Df(x).

Answer 1

You expand the quadratic to get \begin{eqnarray} f(x+h) &=& a (\langle x-x_0+h, x-x_0+h \rangle + \langle b, x+h \rangle \\ &=& f(x) +2a \langle x-x_0, h \rangle + \langle b, h \rangle+ \langle h, h \rangle \end{eqnarray} Dropping the $\|h\|^2$ term we get $Df(x)h = \langle 2a(x-x_0)+b, h \rangle = (2a(x-x_0)+b)^T h$.

Answer 2

My answer is the following (I'm not sure if my answer is correct):

From the definition of Frechet derivative:

$Df(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$

$Df(x)=\lim_{h \to 0}\frac{a\|(x+h)-x^{o}\|^{2}+\langle b,x+h \rangle - a\|x-x^{o}\|^{2}-\langle b,x \rangle}{h}$

$Df(x)=\lim_{h \to 0}\frac{a\|(x+h)-x^{o}\|^{2}+\langle b,h \rangle - a\|x-x^{o}\|^{2}}{h}$

If we consider the euclidean norm, we have:

$Df(x)=\lim_{h \to 0}\frac{a[(x+h)-x^{o}]+\langle b,h \rangle - a[x-x^{o}]}{h}$

$Df(x)=\lim_{h \to 0}\frac{a[h]+\langle b,h \rangle}{h}$

$Df(x)= a+b$