Question

what facts about continuous functions should be proved in order to demonstrate that c œa; b is indeed a subspace as claimed? (these facts are usually discussed in a calcu- lus class.)

Answer 1

To demonstrate that ( Csubseteq [a, b] ) is a subspace, it is essential to prove that the set is closed under addition and scalar multiplication.

Step-by-step explanation:

To establish ( C subseteq [a, b] ) as a subspace, it must satisfy two key conditions: closure under addition and closure under scalar multiplication.Firstly, to show closure under addition, let ( f, g in C). We need to prove that ( f + g in C ). Since ( f ) and ( g ) are continuous functions on ( [a, b] ), their sum( f + g ) is also continuous on ( [a, b] ) by the properties of continuous functions. Therefore,( C ) is closed under addition.

Secondly, for closure under scalar multiplication, let ( f in C ) and ( c ) be a scalar. We need to show that ( cf in C ). Again, as ( f ) is continuous on ( [a, b] ), the product ( cf ) is continuous as well. Thus, ( C ) is closed under scalar multiplication.In summary, by proving closure under addition and scalar multiplication, we establish that ( C subseteq [a, b] ) is a subspace. These properties are fundamental in calculus and ensure that the set ( C ) behaves as a vector subspace within the given interval ( [a, b] ).

Answer 2

To demonstrate that the set c œa; b is a subspace, we need to prove that it satisfies the three subspace properties: closure under addition, closure under scalar multiplication, and contains the zero vector. In the case of continuous functions, the function y(x) must be continuous, and the first derivative of y(x) with respect to x, dy(x)/dx, must be continuous, unless V(x) = ∞.

Step-by-step explanation:

To demonstrate that the set c œa; b is a subspace, we need to prove that it satisfies the three subspace properties: closure under addition, closure under scalar multiplication, and contains the zero vector. In the case of continuous functions, the following facts should be proved:

1. The function y(x) must be continuous.
2. The first derivative of y(x) with respect to x, dy(x)/dx, must be continuous, unless V(x) = ∞.