Q:

Under what circumstances does $\lvert \lambda\rvert \lvert \lvert x \rvert \rvert = \lvert \lvert y \rvert \rvert $ imply $\lambda x = y$

In a proof I saw, we made use of the fact that
for some $y = \lambda_{1}x_{1} + \lambda_{2}x_{2}$ , if we have
$\lvert \lambda_{1}\rvert \lvert \lvert x_{1} \rvert \rvert = \lvert \lvert y \rvert \rvert $, then we can conclude that $y = \lambda_{1}x_{1}$.
This can certainly not be true in a general case, right? What assumptions are needed for it to be true. In the proof I mentioned, the space we were investigating is a Hilbert space.
Edit:
Let $\mathcal{H}$ be a Hilbert space, and $F_{1},F_{2}$ two bounded linear functionals such that $F_{1}\neq 0$ and $F_{2}\neq 0$. Suppose that
$\forall x \in \mathcal{H}: \lvert F_{1}(x)\rvert=\lvert \lvert F_{1}\rvert \rvert \cdot \lvert\lvert x \rvert \rvert \implies  F_{2}(x)=0$
Now show that
$\forall x \in \mathcal{H}: \lvert F_{2}(x)\rvert=\lvert \lvert F_{2}\rvert \rvert \cdot \lvert\lvert x \rvert \rvert \implies  F_{1}(x)=0$
Proof:
Identify, $y_{1},y_{2}$ with $F_{1}(\cdot)=\langle y_{1}, \cdot\rangle$ and $F_{2}(\cdot)=\langle y_{2}, \cdot\rangle$ by Riesz representation, then we can clearly see that:
$\lvert F_{1}(y_{1})\rvert=\langle y_{1},y_{1}\rangle = \lvert \lvert y_{1}\rvert \rvert^{2}\implies \langle y_{2},y_{1}\rangle = 0 $
Now consider the closed subspace $K:=\operatorname{span}\{y_{1},y_{2}\}$. Then, by the orthogonal projection theorem, every $x \in \mathcal{H}$ can be written as $x = \alpha_{1}y_{1}+\alpha_{2}y_{2}+k$ where $k \in K^{\perp}$
And hence we assume that for some $x \in \mathcal{H}$ that $\lvert \langle y_{2}, x\rangle \rvert= \lvert \lvert y_{2}\rvert \rvert \cdot \lvert \lvert x \rvert \rvert$. Note that
$\langle y_{2},x\rangle = \lvert \alpha_{2} \rvert \cdot \lvert \lvert y_{2}\rvert \rvert^{2}\implies  \lvert \alpha_{2} \rvert \cdot \lvert \lvert y_{2}\rvert \rvert^{2}=\lvert \lvert x \rvert \rvert \cdot \lvert \lvert y_{2}\rvert \rvert\implies  \lvert \lvert x \rvert \rvert =\lvert \alpha \rvert \cdot \lvert \lvert y_{2}\rvert \rvert$
And then the implication which I do not understand is stated:
$x = \alpha_{2}y_{2}$, hence implying that $\langle y_{1},x\rangle = 0$.

A:

Note that $y_1, y_2, k$ are pairwise orthogonal, so by Pythagoras:
$||x||^2=||\alpha_1y_1||^2+||\alpha_2y_2||^2+||k||^2$
But during the proof we also got that $||x||^2=||\alpha_2y_2||^2$. Hence we must have $||\alpha_1y_1||^2+||k||^2=0$, which implies $\alpha_1y_1=k=0$.

