If such an $f$ existed, boundedness would make its isolated singularity at zero removable. Let $F$ be the holomorphic extension to the disc. Continuity gives $1\le |F(0)|\le 2$. Equality at either endpoint contradicts the minimum-modulus principle applied to $F$ or the maximum modulus principle, since $F$ is nonconstant. Hence $F(0)\in A$. Surjectivity of $f$ supplies some nonzero $z$ with $f(z)=F(0)$, contradicting injectivity of the extension. Therefore

The closure $\overline{Z}$ is the intersection of all closed subsets containing $Z$; equivalently, every neighbourhood of a point of $\overline{Z}$ meets $Z$. The subspace is dense when $\overline{Z}=X$. A space is Hausdorff when distinct points have disjoint neighbourhoods.

For Hausdorff $Y$, the diagonal ${\Delta }_Y$ is closed in $Y\times Y$. Thusis closed. Since it contains the dense set $Z$, it equals $X$.

The conclusion fails without Hausdorffness. Let $X=Y=\{0,1\}$ have the Sierpinski space topology $\{\varnothing ,\{1\},X\}$ and let $Z=\{1\}$, which is dense. The identity map and the constant map with value $1$ are continuous and agree on $Z$, but differ at $0$.

A Hermitian form is conjugate-linear in its first argument, linear in its second, and satisfies $\varphi (w,v)=\overline{\varphi (v,w)}$. For $H=\varphi -\psi$, the hypothesis says $H(v,v)=0$. The complex polarization identitytherefore gives $H(v,w)=0$ for all $v,w$.

The displayed matrix is Hermitian. By Sylvester's criterion, its leading principal minors areHence the form is

Irrotationality gives ${\mathbf{u}}_i=\nabla {\varphi }_i$ and ${\nabla }^2{\varphi }_i=0$. At $z=\zeta$, the exact kinematic conditions areand pressure continuity gives $p_1=p_2$. Bernoulli equation in each fluid is

Linearize at $z=0$ and write $\zeta =Z{e}^{i(kx-\omega t)}$. Decay at infinity givesThe kinematic conditions give $-i\omega Z=-k{A}_1=k{A}_2$, while pressure continuity givesElimination yieldsStable waves require the lower fluid to be denser, ${\rho }_2>{\rho }_1$; otherwise this is the Rayleigh-Taylor instability.