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Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite I encountered the problem of trying to find the examples of complex varieties which are not the complexification of a real variety, when I considered the Exercise II.4.7.(a) in Hartshorne's book. I wonder whether there are indeed some interesting complex varieties which don't have the so called semi-linear involution in Hartshorne's book. For rational affine curve, I have found the following example $$A=mathbbC[x,frac1x-1,frac1x-i,frac1x-i-1]$$ by a little hard calculation, considering the possible semi-linear involutions of the field $mathbbCleft(xright)$ induced by the complex conjugation of $mathbbC$ . But I really wonder whether there are some interesting examples of complex projective varieties without real descent. I have poor knowledge on the classical topics in algebraic geometry, therefore I find it difficult to work out the problem by myself. Could anyone shed some light on my confusion? Th...

Clash Royale CLAN TAG #URR8PPP up vote 6 down vote favorite 1 Let $E$ a vector space of infinite dimension and let $(e_i)_imathbb N$ a basis. How is interpreted $$sum_iinmathbb Nx_ie_i ?$$ Suppose now that $E$ is a Banach space with norm $|cdot |$. I suppose that $$sum_iinmathbb Nx_ie_i=lim_nto infty sum_i=1^n x_ie_i,$$ in the $|cdot |$ sense, i.e. $$forall varepsilon>0, exists N: forall nin mathbb N, ngeq Nimplies left|sum_iinmathbb Nx_ie_i-sum_i=1^nx_ie_iright|<varepsilon.$$ But since $E$ is a vector space, $sum_iinmathbb Nx_ie_iin E$ and thus exist (by definition of a vector space), but I guess it could happen that $$lim_nto infty sum_i=1^n x_ie_i$$ doesn't exist or is infinite, does it ? So how can we manage this case ? linear-algebra functional-analysis vector-spaces summation share | cite | improve this question edited Jul 25 at 10:34 Sou 2,706 2 8 20 asked Jul 25 at 10:26 user330587 821 3 10 add a comment Â...