Good day, I am confused with it: the application $x\rightarrow (x,y)$ is uniform continuous for $y$ in a Hilbert space.
I see that $\left | (x_{1},y)-(x_{2},y) \right |=\left | (x_{1}-x_{2},y) \right |\leq \left \| x_{1}-x_{2} \right \|\left \| y \right \|$.
My question is how understand it with the definition:$∀ε>0:∃δ>0:∀p,q∈X:d_{X}(p,q)<δ⟹d_{Y}(f(p),f(q))<ε$.
If $\boldsymbol y = 0$, then the mapping is constant therefore uniform continuity is trivially verified.
Suppose $\boldsymbol y \neq 0$. Let $\epsilon > 0$ and $\delta = \frac{\epsilon}{||y||}$. Then, using Cauchy-Schwarz inequality, if $p,q$ are such that $||p - q|| < \delta$, we have that:
\begin{align*} |\langle p, y \rangle - \langle q, y \rangle| \leq ||p - q||\cdot||y||<\frac{\epsilon}{||y||}\cdot ||y|| = \epsilon \end{align*}
Therefore, $x \mapsto \langle x, y \rangle$ is uniformly continuous.