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Let $X$ be a Banach space such that both $X$ and $X^*$ have finite cotype. Also assume that $X$ is an inductive limit of finite dimensional Banach spaces $X_n\subseteq X_{n+1}.$ Fix $1<p<\infty.$ Is there any known result which can give precise information about finite dimensional subspaces $Y_n$'s of $X$ with $\sup\limits_n\text{dim}Y_n=\infty$, $Y_n$'s have Banach-Mazur distance $\lambda_n$ from $\ell_p^n$'s and the projection constants from $X$ to $Y_n$ have some bound, say $P_n.$ I want best possible choice of $Y_n$'s so that the bounds of $\lambda_n$ and $P_n$ in terms of $n$ and $p$ and best possible.

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You can find many relevant results in the book: Pisier, Gilles Factorization of linear operators and geometry of Banach spaces. CBMS Regional Conference Series in Mathematics, 60. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. See, for example, Theorem 10.6 and Corollary 10.8.

The mentioned book can be used as a good starting point for learning numerous known results related to your question.

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