Doubt about how to find a Lipschitz constant

Question

I have a doubt about a sentence of my Calculus text.

Let $f: [t_1, t_2]\times \mathbb{R}^n \to \mathbb{R}^n, (t,y)\to f(t,y)$ such that $|\partial_{y_j} f_i|$ is continuous and bounded for every $i,j=1,...n$. Then f has $\sqrt n L$ as Lipschitz constant (with respect to y uniformly in t) where $L>0$ is such that $|\partial_{y_j} f_i|\le L$.

I don't know how to get $\sqrt n L$ as Lipschitz constant:
for $i=1,...,n$ we have $|f_i(t,y)-f_i(t,z)|\le|\nabla f_i(t, \theta y)||y-z|$ for some $\theta\in[0,1]$; and since $|\nabla f_i(t, \theta y)|\le \sqrt{nL^2} $ we obtain $|\nabla f(t, \theta y)|\le \sqrt{n^2L^2}=nL$.
Do you know ways to improve my inequality? Thanks in advance.

Answer 1

Indeed, I think that there is an error in the book. That the example of $n=2$ and $f$ the linear map with all coefficients equal to $1$. Then the $\partial_j f_i$ are also all equal to one. Take $x=(0,0)$ and $y=(1,1)$. You have $\Vert f(x)-f(y) \Vert_2 = 2\sqrt{2} =2 \Vert x-y\Vert_2 $.

Answer 2

I suspect it is using the fact that $\|x\|_2 \le \|x\|_1 \le \sqrt{n} \|x\|_2$.

Consider the path $x_0=x \to (y_1,x_2,...,x_n) \to (y_1,y_2,x_3,...,x_n) \to \cdots \to x_n=y$, where one component changes at a time. \begin{eqnarray} \|f(x,t)-f(y,t)\|_2 &\le& \sum_k \|f_(x_{k+1},t)-f(x_k,t)\|_2 \\ &\le& \sum_k L \|x_{k+1}-x_k\|_2 \\ &=& L \sum_k |y_k-x_k| \\ &=& L \|x-y\|_1 \\ &\le& \sqrt{n}L \|x-y\|_2 \end{eqnarray}