Show L(X,Y) is a vector space. Then if X,Y are n.l.s. over the same scalar field define B(X,Y) = set of all bounded linear operators for X and Y
Show B(X,Y) is a vector space(actually a subspace of L(X,Y)
The Attempt at a Solution
im not sure if i have taken this question down properly.
To prove some set is a vector space you have to show the 4 axiom of a vector space hold. namely-for u, v, w be arbitrary vectors in V, and a, b be scalars in F
1. u + (v + w) = (u + v) + w.
2. v + w = w + v.
3. There exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V.
Inverse elements of addition For all v ∈ V, there exists an element w ∈ V, called the additive inverse of v, such that v + w = 0.
4. a(v + w) = av + aw
Im not sure how to progress with this