Let $H$ be Hilbert space with inner product $(\cdot, \cdot)$ and let $H_1$ and $H_2$ be two finite-dimensional subspaces of $H$ such that $H_1 \cap H_2 = \{0_H\}$, where $0_H$ is the zero of $H$. Let then $H_s = H_1 \oplus H_2$ be the direct sum of $H_1$ and $H_2$. It is possible to prove (see, e.g., [Theorem 1, 1]) that for any $v_1\in H_1$ and $v_2 \in H_2$ there exists $\gamma \in [0, 1)$ such that $$ (v_1, v_2)\leq\gamma \|v_1\|\, \|v_2\|. $$ Now let $w \in H_s$ and let $w = w_1 + w_2$ with $w_1 \in H_1$, $w_1 \neq 0_H$ and $w_2 \in H_2$. I have the intuition that there exists $\lambda \in [0, 1)$ (related to $\gamma$) such that for all $v_2 \in H_2$ it holds $$ (w, v_2)\leq \lambda \|w\| \, \|v_2\|. $$ Is there an easy way to prove this?
Edit:
Applying the strengthened Cauchy-Schwarz inequality between $H_1$ and $H_2$ one gets \begin{align} (w,v_2) &= (w_1, v_2) + (w_2, v_2) \\ &\leq \gamma \|w_1\|\,\|v_2\| + \|w_2\|\,\|v_2\| \\ &= (\gamma \|w_1\| + \|w_2\|) \|v_2\|. \end{align} Now everything would be fine if there existed a constant $0 \leq \lambda < 1$ such that $$ \gamma \|w_1\| + \|w_2\| \leq \lambda \|w\|. $$
Edit 2:
I kind of realized that it will not be possible to obtain a $\lambda$ for any choice of $w \in H_s$. Nevertheless, is it possible to obtain an estimate depending on the ratio between the norms $\|w_1\|$ and $\|w_2\|$ of its components in $H_1$ and $H_2$, or on the quantity $\|w_1 - w_2\|$?
Yes. Let $H’_1$ be a subspace of $H$ spanned by $w$. Then $H’_1\cap H_2=\{0_H\}$, so the same theorem implies the existence of the required $\lambda$.