By Chambert-Loir A.

This graduate path has faces: algebra and geometry. certainly, we research at the same time loci of issues outlined by way of polynomial equations and algebras of finite kind over a box. we will exhibit on examples (Hilbert's Nullstellensatz, measurement idea, regularity) how those are faces of a unmarried head and the way either geometrie and algebraic features enlight the single the opposite.

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**Example text**

Now t h e disked h u l l A 1 o f A i s contained i n M t $ V , f o r $V i s d i s k e d . Thus: and t h e a s s e r t i o n i s proved. (e) : I n every topological vector space the precompact bornology i s f i n e r than t h e von Neumann bornology. T h i s i s an immedia t e consequence o f t h e d e f i n i t i o n s . EXAMPLE ( 1 0 ) : The Compact Bornology o f a Banach Space: We have s t a t e d e a r l i e r t h a t t h e compact bornology o f a l o c a l l y convex s p a c e , even normed, i s n o t convex i n g e n e r a l ( f o r a counter-exanple, s e e E x e r c i s e 4 .

I f V €9,t h e r e exi s t s W e v such t h a t W t W C V . By v i r t u e of t h e e q u i c o n t i n u i t y o f H I and H 2 , t h e s e t s H l - l ( W ) and H 2 - I ( W ) a r e neighbourhoods of zero i n E . Now ( H I t H 2 ) - l ( V ) c o n t a i n s H 1 - 1 ( W ) n H 2 - 1 ( W ) and hence i s a neighbourhood o f zero i n E . Thus H I t H 2 e x s i n c e V was a r b i t r a r y i n 0. The family i s a l s o s t a b l e under homothetic t r a n s f o r m a t i o n s s i n c e f o r kvery H e x , X e x and Ve’B w e have : EXAMPLE n x which shows t h e s e t ( X H l - 1 t o be a neighbourhood o f zero i n E .

For every s c a l a r A e M , t h e map x -+ Ax o f E i n t o E i s continuous, hence f o r every compact s e t A C E, AA i s compact. S i m i l a r l y , t h e c o n t i n u i t y o f t h e map ( x , y ) x t y o f E x E i n t o E ensures t h a t t h e s e t A t B i s compact whenever A and B a r e compact s u b s e t s o f E. F i n a l l y , f o r e v e r y compact A C E, t h e c i r c l e d h u l l o f A i s compact, s i n c e i t i s t h e image o f t h e s e t D x A ( D t h e compact u n i t b a l l o f M ) under t h e continuous map (X,x> Ax o f M x E i n t o E .

### Algebre commutative et introduction a geometrie algebrique by Chambert-Loir A.

by Thomas

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