Question

In: Advanced Math

Let C(R) be the vector space of continuous functions from R to R with the usual...

Let C(R) be the vector space of continuous functions from R to R with the usual addition and scalar multiplication. Determine if W is a subspace of C(R). Show algebraically and explain your answers thoroughly.

a. W = C^n(R) = { f ∈ C(R) | f has a continuous nth derivative}

b. W = {f ∈ C^2(R) | f''(x) + f(x) = 0}

c. W = {f ∈ C(R) | f(-x) = f(x)}.

Expert Solution

a). { has a continuous nth derivative } .

Let , .

are continuous , where denotes nth derivative of .

Now , are also contiunuous as sum of two contunuous function are also continuous .

So W is closed under addition .

is also continuous

So W is closed under scalar multiplication .

Hence W is a subspace.

.

(b). { }

Let ,

Now ,

Also ,

So

Hence W is a subspace .

.

(c). { }

Let ,

Also

So W is closed under addition and salar multiplication . Hence W is subspace .

Colby Messinger answered 2 years ago

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