I was reading Hilbert space chapter In that I came across following. I can follow proof given in book.But I wanted to know what is geometric meaning of
$$\langle f-u,v-u \rangle \leq 0,\forall v\in K$$
Also there is remark given below proof.In that author says $f:[0,1]\to R$ achieves minimum at 0 then $f'(0)\leq 0$ but it should be $f'(0)\geq 0$ also $f'(1)\leq 0$ but auther says $f'(1)=1$ I do not understand why?
Please, anyone, help me understand better.
Thanking you
Regarding your primary question: note that $\cos^{-1}(\langle f-u,v - u \rangle)$ is the angle between $f-u$ and $v - u$. Stating that $\langle f-u,v - u \rangle\leq 0$ for all $v$ is equivalent to stating that the angle between $f-u$ and $v - u$ is always either right or obtuse.
To get a feeling for why this should be the case, I would suggest that you consider the simple example of $$ H = \Bbb R^2, \quad f = (2,0), \quad K = \{(x,y): x^2 + y^2 \leq 1\}. $$ In this case, we find that $u = (1,0)$. You should find $f$ and $K$ easy enough to draw. Choose a random $v$ (or several), and consider what the vectors $f-u,v-u$ look like.