Exercise:
Let varnothing neq S subseteq V. Then textSpSa_v_+...+a_nv_n|nin mathbbN a_iin K v_iin S forall leq ileq n textsubset of W obtained by taking all possible linear combinations of vectors from S Reminder: The set S can be finite or infinite. Regardless of that each linear combination uses only a finite number of elements from S.

Solution:
Proof. Denote by overlinetextSpSa_v_+...+a_nv_n|nin mathbbN a_iin K v_iin S forall leq i leq n. We'll first show that: abcliste abc S subseteq overlinetextSpS. abc overlinetextSpS subseteq V is a linear subspace. abc forall subspaces Wsubseteq V with Ssubseteq W we have overlinetextSpSsubseteq W. abcliste Note that a+bLongrightarrow textSpSsubseteq overlinetextSpS by the previous Lemma and cLongrightarrow overlinetextSpS subseteq bigcap_W in mathcalN W textSpS with mathcalNW|Ssubseteq W Wsubseteq V is a subspace. So together we have overlinetextSpStextSpS which is what we wanted. Now we continue by proving a-c. abcliste abc If vin S Longrightarrow v v in overlinetextSpS. abc We need to show that in overlinetextSpS and that forall vwin overlinetextSpS forall alpha beta in K: alpha v+beta w in overlinetextSpS. Indeed Sneq varnothing by asption. Pick any u in S. Then uin overlinetextSpS. Let vwin overlinetextSpS alpha betain K. Write va_v_+...+a_nv_n wb_w_+...+b_mw_m where nmin mathbbN a_ib_j in K v_iw_j in S forall leq i leq n leq jleq m. Now alpha v+beta w alpha a_v_+...+alpha a_nv_n+beta b_w_+...+beta b_mw_m which is a linear combination of the vectors v_...v_nw_...w_min S with coeffs alpha a_...alpha a_nbeta b_...beta b_m in K. This proves b. abc Let Wsubseteq V be a linear subspace s.t. S subseteq W. Let vin overlinetextSpS. We need to show that vin W. Indeed write va_v_+...+a_nv_n for some nin mathbbN v_iin S a_i in K. By asption S subseteq W hence v_iin W forall i. Since W is a linear subspace we have va_v_+...+a_nv_n in W. abcliste