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Val Suran From Wikipedia, the free encyclopedia Jump to navigation Jump to search Commune in Bourgogne-Franche-Comté, France Val Suran Commune Location of Val Suran Val Suran Show map of France Val Suran Show map of Bourgogne-Franche-Comté Coordinates: 46°23′42″N 5°27′14″E  /  46.395°N 5.454°E  / 46.395; 5.454 Coordinates: 46°23′42″N 5°27′14″E  /  46.395°N 5.454°E  / 46.395; 5.454 Country France Region Bourgogne-Franche-Comté Department Jura Arrondissement Lons-le-Saunier Canton Saint-Amour Intercommunality Petite Montagne Area 1 37.49 km 2 (14.47 sq mi) Population (2013) 2 831  • Density 22/km 2 (57/sq mi) Time zone UTC+01:00 (CET)  • Summer (DST) UTC+02:00 (CEST) INSEE/Postal code 39485 /39320 1 French Land Register data, which excludes lakes, ponds, glaciers > 1 km 2 (0.386 sq mi or 247 acres) and river estuaries. 2 Pop

4 2 $begingroup$ I would like to show the following: Given a vector spaces $V$, a subspace $S subset V$ and an the dual space $V^*$ to $V$. Show that: $$dim(N)+dim(S) = dim(V) = dim(V^*)$$, where $N subset V^*$ is the annihilator to $S$. Assume only finite dimensional vector spaces. I tried a proof but I am not sure if this is correct: Proof: I give a try :-) Assume that $dim(S) = m$, $dim(V) = dim(V^*) = n$ The annihilator is given as: $$N:= { boldsymbol{alpha} mid langle boldsymbol{alpha},mathbf{x} rangle =0 quad ,quad forall mathbf{x} in S }$$ Where $ langle boldsymbol{alpha},x rangle$ is the duality pairing, it is a bilinear form as: $$ begin{aligned} textrm{B}( boldsymbol{alpha},mathbf{x} ) : V^* times V &rightarrow mathbb{R} \ boldsymbol{alpha},mathbf{x} &mapsto langle boldsymbol{a