Let $V$ be a vector space with a basis $f_1, \ldots, f_n$. Here $f_i$ are complex functions on $\mathbb{C}$. Defined the sequence of exponents of $V$ at $\lambda \in \mathbb{C}$ as follows. There is a basis $g_1, \ldots, g_n$ of $V$ and a sequence of integers $c_1<\cdots
sequence of exponents of a vector space at a point
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complex-analysis
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0What have you tried? If you have a different basis $(h_i)_i$ and integers d_1<\cdots
that do the same thing for $h$ as the $c$'s for the $g$'s, then you must show that $d_1=c_1$, $d_2=c_2$, etc. Perhaps by induction on $n$? (Note without extra assumptions -- such that all functions are holomorphic in a neighborhood of $\lambda$ -- it would be possible that there is no matching $c_i$ sequence at all, except that the exercise specifies as a premise that there is at least one). – 2011-09-28