Is this a typo in an exercise about Hilbert spaces?

Question

I'm doing an exercise about functional analysis:

My questions:

1. In question 5,

For every $x \in H,$ justify the existence of a unique vector $T_{x} \in H$ such that for all $y \in K$ $$ \left\|x-T_{x}\right\| \leq\|x-y\| $$

Should it be "a unique vector $T_{x} \in K$" rather than "a unique vector $T_{x} \in H$"? In this way, we can apply orthogonal projection on a closed convex set $K$. In the original question, clearly $T_x=x$ satisfies. If $z$ is close to $x$ enough, then $T_x = z$ also satisfies.

Could you please verify if my observation is correct?

2. Please shed me some lights on computing $T_x$ in question 7 and proving $T_{-x}=-T_{x}$ in question 8!

Thank you so much for your help!