Prove that the set V of all real valued continuous (differentiable or integrable) functions defined on the closed interval [a, b] is a real vector space with the vector addition and scalar multiplication defined as follows: (f +g)(x)=f(x) + g(x) (λ f)(x)=λ f(x) for all f, g ∈V and λ ∈R.
Vector Arithmetic
Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.
Vector Calculus
Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.
Prove that the set V of all real valued continuous (differentiable or integrable) functions defined on the closed interval [a, b] is a real
(f +g)(x)=f(x) + g(x)
(λ f)(x)=λ f(x) for all f, g ∈V and λ ∈R.
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