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"=" is preferred over ":=" in definition. There is also no need to define the limit as f when the function is never referred to. Tags: Mobile edit Mobile web edit |
→Properties: There's no need to have a long parenthetical referring to a definition when there is a link to the page for semi-continuity. Tags: Mobile edit Mobile web edit |
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*If <math>f_1,f_2,\dots</math> is a decreasing sequence of plurisubharmonic functions then its pointwise limit is plurisubharmonic.
*Every continuous plurisubharmonic function can be obtained as the limit of a monotonically decreasing sequence of smooth plurisubharmonic functions. Moreover, this sequence can be chosen uniformly convergent.<ref>R. E. Greene and H. Wu, ''<math>C^\infty</math>-approximations of convex, subharmonic, and plurisubharmonic functions'', Ann. Scient. Ec. Norm. Sup. 12 (1979), 47–84.</ref>
*The inequality in the usual [[semi-continuity]] condition holds as equality, i.e. if <math>f</math> is plurisubharmonic then <math>\limsup_{x\to x_0}f(x) =f(x_0)</math>.
* Plurisubharmonic functions are [[Subharmonic function|subharmonic]], for any [[Kähler manifold|Kähler metric]].
*Therefore, plurisubharmonic functions satisfy the [[maximum principle]], i.e. if <math>f</math> is plurisubharmonic on the [[connected space|connected]] open ___domain <math>D</math> and
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