What does the space $\overline{\mathbb{C}[z]}$ stands for? Does it contain all the analytic functions or there are something else? And what about the closure thing?
The space $\mathbb{C}[z]$
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$\begingroup$
complex-analysis
notation
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6It is impossible to know what that notation means if you do not tell us at least where you found it... – 2011-11-22
1 Answers
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$\mathbb{C}[z]$ is the space of polynomial functions. I'm not sure what the bar does to it. What context does this appear in? Perhaps $\overline{\mathbb{C}[z]}$ stands for the space of entire functions because they are the limits of polynomials since they have series expansions. Other notations for the space of entire functions are $\mathcal O(\mathbb C)$ and $\mathcal H(\mathbb C)$.
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2I know that's what you meant - trouble is, neither one of us knows what the author meant. All I'm saying is that if *I* were to use that symbol, I would mean for it to include anything that looked like a limit of a sequence of polynomials. I might even mean for it to include things like $1+x+2x^2+6x^3+24x^4+120x^5+\dots$, which doesn't converge anywhere but at zero. – 2011-11-22