Real Algebraic Geometry II – Exercise Sheet 6

Universität Konstanz Tom-Lukas Kriel Department of Mathematics and Statistics María López Quijorna

Summer Term 2017 Markus Schweighofer

Real Algebraic Geometry II – Exercise Sheet 6

Exercise 1 (8P) Let K be a subfield of R and V be a K-vector space. We call a map p: V

→

_R≥0aseminormon Vif

∀

x,y

∈

V : p

(

x

+

y

) ≤

p

(

x

) +

p

(

y

)

and

∀

λ

∈

K :

∀

x

∈

V : p

(

λx

) = |

λ

|

p

(

x

)

. Now letV be a topologicalK-vector space.

(a) Show that any neighborhoodUof 0 inVisabsorbing, i.e., V

=

^[

λ∈K>0

λU.

(b) Show that for any convex balanced neighborhoodUof 0 inV, the function p_U: V

→

_R, x

7→

inf

{

λ

∈

K>0

|

x

∈

λU

}

is a seminorm onVwith

U^◦

= {

x

∈

V

|

p

(

x

) <

1

} ⊆

U

⊆ {

x

∈

V

|

p

(

x

) ≤

1

} =

U.

(c) Prove that the correspondence

U

7→

p_U

{

x

∈

V

|

p

(

x

) <

1

} ←

_[p

defines a bijection between the set of all open convex balanced neighborhoods of 0 inVand the set of all continuous seminorms onV.

Exercise 2 (6P) Let K be a subfield of R and V be a K-vector space. If P is a set of seminorms onV, we denote byOP the topology onV generated by the sets

{

x

∈

V

|

p

(

x

−

y

) <

ε

} (

p

∈

P,y

∈

V,ε

>

0

)

.

We call P separatingif for all x

∈

V

\ {

0

}

there exists p

∈

P such that p

(

x

) 6=

0. Now letO be a topology on the setV. Show that the following are equivalent:

(a)

(

V,O

)

is a locally convexK-vector space.

(b) There exists a separating set Pof seminorms onV such thatO

=

_OP.

Exercise 3(6P) Let n

∈

_N≥2. The one-dimensional affine subspaces ofRⁿ are called lines.

(a) Show that the following defines a topology onRⁿ: A set A

⊆

_Rⁿ is open if and only if for every line G the intersection G

∩

A is open in G with respect to the topology induced onGbyRⁿ.

(b) Is the additionRⁿ

×

_Rⁿ

→

_Rⁿ,

(

x,y

) 7→

x

+

y continuous with respect to this topology?

(c) Is the scalar multiplicationR

×

_Rⁿ

→

_Rⁿ,

(

λ,x

) 7→

λx continuous with respect to this topology?

Exercise 4 (8P+2BP) This exercise should be done without using the separation theo- rems in §7 of the lecture notes. Let Abe a nonempty closed convex subset ofRⁿ.

(a) Show that for eachx

∈

_Rⁿ, there is a uniqueπ

(

x

)

:

=

yinRⁿsuch that

k

x

−

y

k < k

x

−

z

k

for allz

∈

A

\ {

x

}

.

(b) Show that the corresponding mapπ: Rⁿ

→

Ais contractive, i.e.,

k

π

(

x

) −

π

(

y

)k ≤ k

x

−

y

k

for allx,y

∈

_Rⁿ.

(c) LetA

⊆

_Rⁿbe closed and convex andx

∈

_Rⁿ

\

A. Show that there is anR-linear ϕ:Rⁿ

→

_Rwith ϕ

(

x

) <

ϕ

(

a

)

for all a

∈

A.

(d) (Bonus) LetC

⊆

_Rⁿ be compact and convex. Let A

⊆

_Rⁿbe closed and convex such thatA

∩

C

=

_∅. Prove: There areε

>

0 and anR-linear function ϕ: Rⁿ

→

_R such thatϕ

(

x

) +

ε

≤

ϕ

(

a

)

for all a

∈

Aandx

∈

C.

Please submit until Tuesday, June 6, 2017, 9:55 in the box named RAG II near to the room F411.