Let $\mathbb{R}^{n\times n}$ be the vector space of square matrices with real entries. For each $A\in \mathbb{R}^{n\times n}$ we consider the norms given by: $ \displaystyle\|A\|_1=\max_{1\leq j\leq n}\sum_{i=1}^{n}|a_{ij}|; $ $ \displaystyle\|A\|_\infty=\max_{1\leq i\leq n}\sum_{j=1}^{n}|a_{ij}|; $
$ \displaystyle\|A\|_\text{max}=\max\{|a_{ij}|\}. $ Matrix $A\in \mathbb{R}^{n\times n}$ is said to be positive definite iff $ \langle Ax, x\rangle> 0 \quad \forall x\in\mathbb{R}^n\setminus\{0\}. $ Let $S$ be the set of all positive definite matrices on $\mathbb{R}^{n\times n}$. Prove that $S$ is an open set in $(X,\|.\|_1)$, $(X,\|.\|_\infty$), $(X,\|.\|_\text{max})$.
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