Let (V, ||·||) be a normed space, and W a
dNormV,||·|| -closed vector subspace of
V.
(a) Prove that a function |||·||| : V /W → R≥0 can be
consistently defined by ∀v ∈ V : |||v + W||| df= inf({||v + w|| :
R≥0 | w ∈ W}).
(b) Prove that |||·||| is a norm on V /W.
(c) Prove that if (V, ||·||) is a Banach space, then so is (V
/W, |||·|||)
4. Let n ≥ 8 be an even integer and let k be an integer with 2 ≤
k ≤ n/2. Consider k-element subsets of the set S = {1, 2, . . . ,
n}. How many such subsets contain at least two even numbers?
Let N(n) be the number of all partitions of [n] with no
singleton blocks. And let A(n) be the number of all partitions of
[n] with at least one singleton block. Prove that for all n ≥ 1,
N(n+1) = A(n). Hint: try to give (even an informal) bijective
argument.
For any n ≥ 1 let Kn,n be the complete bipartite graph (V, E)
where V = {xi : 1 ≤ i ≤ n} ∪ {yi : 1 ≤ i ≤ n} E = {{xi , yj} : 1 ≤
i ≤ n, 1 ≤ j ≤ n} (a) Prove that Kn,n is connected for all n ≤ 1.
(b) For any n ≥ 3 find two subsets of edges E 0 ⊆ E and E 00 ⊆ E
such...
Let U and V be vector spaces, and let L(V,U) be the set of all
linear transformations from V to U. Let T_1 and T_2 be in
L(V,U),v be in V, and x a real number. Define
vector addition in L(V,U) by
(T_1+T_2)(v)=T_1(v)+T_2(v)
, and define scalar multiplication of linear maps as
(xT)(v)=xT(v). Show that under
these operations, L(V,U) is a vector space.
Let u and v be two integers and let us assume u^2 + uv +v^2 is
divisible by 9. Show that then u and v are divisible by 3. (please
do this by contrapositive).