Bollob\'{a}s-type inequalities for subspaces via weight invariance

Let $V$ be an $n$-dimension real vector space with a direct sum decomposition $V = V_1 \oplus \cdots \oplus V_r$. Let $\mathcal{P} = \{(A_i, B_i) : i \in [m]\}$ be a skew Bollob\'as system of subspaces of $V$ such that each $i\in [m]$, $A_i = \bigoplus_{k=1}^r (A_i \cap V_k)$ and $B_i = \bigoplus_{k=1}^r (B_i \cap V_k)$. We prove that $$\sum_{i=1}^{m} \prod_{k=1}^{r} \left[ \binom{a_{i,k} + b_{i,k}}{a_{i,k}} (1 + a_{i,k} + b_{i,k})^{-1} \right] \leq 1,$$ where $a_{i,k} = \dim(A_i \cap V_k)$ and $b_{i,k} = \dim(B_i \cap V_k)$. This extends a recent result of Yue from set systems to finite dimensional subspaces. We then consider Tuza's theorem on weak Bollob\'as system for $d$-tuples. We give an alternative proof of the original set version of Tuza, and also establish its vector space analogue. Precisely, let $\mathcal{P} = \{(A_i^{(1)}, \ldots, A_i^{(d)}) : i \in [m]\}$ be a skew Bollob\'as system of $d$-tuples of subspaces of finite dimensional space $V$ with $a^{(\ell)}_i=\dim (A_i^{(\ell)})$. Then, for any positive real numbers $p_1, \ldots, p_d$ satisfying $p_1 + \cdots + p_d = 1$, we prove that $\sum_{i=1}^{m} \prod_{\ell=1}^{d} p_{\ell}^{a_i^{(\ell)}} \leq 1.$