MathLearner123
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I'm reading "From Holomorphic Functions to Complex Manifolds" - Fritzsche & Grauert and I have something that I don't understand very well: If ##\nu \in \mathbb{N}_0^n, t \in \mathbb{R}^n_+## and ##z \in \mathbb{C}^n##, write ##\nu = (\nu_1, \nu'), t = (t_1, t')## and ##z = (z_1, z')##. \ An element ##f=\sum_{\nu \geq 0} a_\nu \mathbf{z}^\nu \in \mathbb{C} [\![ z ]\!]## (formal power series ring) can be written in the form ##f=\sum_{\lambda = 0}^\infty f_\lambda z_1^\lambda## where ##f_\lambda(z')=\sum_{\nu' \ge 0} a_{(\lambda, \nu')} (z')^{\nu'}##. Why can we write this? Changing the order of factors doesn't affect the formal power series? We have that ##f=\sum_{\nu \ge 0} a_\nu z^\nu = \sum_{(\nu_1, \nu')\ge 0} a_{(\nu_1, \nu')}z_1^{\nu_1} (z')^{\nu'} = \sum_{\nu_1 \ge 0} \sum_{\nu' \ge 0} a_{(\nu_1, \nu')}z_1^{\nu_1} (z')^{\nu'} = \sum_{\nu_1 \ge 0} \left(\sum_{\nu' \ge 0} a_{(\nu_1, \nu')} (z')^{\nu'}\right)z_1^{\nu_1} = \sum_{\lambda \ge 0} \left(\sum_{\nu' \ge 0} a_{(\lambda, \nu')} (z')^{\nu'}\right)z_1^{\lambda} = \sum_{\lambda \ge 0} f_\lambda z_1^{\lambda}##. But taking out that common factor ##z_1^{\nu_1}## doesn't change the formal power series? And another thing that I dont't understand: In what order the terms appear in the expansion ##f = \sum_{\nu \ge 0} a_\nu z^\nu##. For example, in ##\mathbb{C}[\![z_1,z_2,z_3]\!]## both terms ##a_{(1,2,3)} z^{(1,2,3)}## and ##a_{(1,3,2)} z^{(1,3,2)}## have "order" ##1+2+3=6## but who comes first in the expansion? I need to consider some order on ##\mathbb{N}_0^3## or something?