Exercise:
Let V be a vector space over K and Usubseteq V a subspace. Define U^bot:lin V^*|l|_U all functionals on V that vanish on U. Then: &U^botsubseteq V^*quad textis a linear subspace. &existsquad texta canonical isomorphism V/U^* cong U^bot If we ase that V is finite dimensional then exists a canonical isomorphism U^* cong V^*/U^bot.

Solution:
Proof. Clearly in U^bot. If l_ l_in U^bot and alpha beta in K then alpha l_+beta l_ in U^bot because forall uin U we have alpha l_+beta l_ualpha l_u+beta l_u. Define a map L:V/U*longrightarrow U^bot as follows: Let Sin V/U^* i.e. S:V/Ulongrightarrow K. Define LSin V^* to be LS: Scirc pi. Put shortly: L is the map Slongmapsto Scirc pi. We claim that L is an isomorphism. To see this define a map P:U^botlongrightarrow V/U^* as follows. Let Tin U^bot i.e. T:Vlongrightarrow K s.t. T|_U. By theorem on the universal property of quotients exists ! T':V/Ulongrightarrow K s.t. TT'circ pi. Define PTT'. We claim that Lcirc Pid_U^bot Pcirc Lid_V/U^*. Indeed forall S:V/U longrightarrow K we have: Pcirc LSPScirc piS. Also forall Tin U^bot Lcirc PTLT'Tcirc piT. This proves Lcirc Pid_U^bot Pcirc Lid_V/U^*Longrightarrow L is injective and surjective hence an isomorphism note that we don't have to check that P is linear since we know that L is. It follows that P is also a linear isomorphism. Ase now that V is finite dimensional. Consider the following map R:V^*longrightarrow U^*: forall phi in V^* i.e. a linear funciton phi:Vlongrightarrow K define Rphi:phi|_Uin U^* phi_U:Ulongrightarrow K the restriction of phi to U. We claim that R is surjective. Indeed let Psiin U^*. We need to show that exists phiin V^* s.t. phi|_UPsi. The finite dimensionality of V is important here. Pick a basis u_...u_k for U and ext it to a basis u_...u_kv_k+...v_n of V. Define a linear phi: Vlongrightarrow K by phiu_i:Psiu_i forall leq ileq k phiv_j: forall k+leq jleq n. Clearly phi|_UPsi because phi and Psi coincide on the elements of a basis of U. This proves textImRU^*. We claim that textKerRU^bot. Indeed this follows directly from definitions: textKerRphiin V^*|phi|_UU^bot. By the isomorphism theorem R induces an isomorphism overlineR:V^*/U^botlongrightarrow U^*