Question

determine which of the following are subspaces of the space F(-infinity,infinity)
(a) all f such that f(x)< or equal to 0 for all x
(b) all f such that f(0)=0
(c) all f such that f(0)=2
(d) all constant functions
(e) all f of the form k1+k2sinx , where k1 and k2 are real numbers

Answer 1

What's F(-infinity,infinity)?

Answer 2

F denotes continuous functions

Answer 3

No D is ok

Answer 4

please explain sir

Answer 5

Never mind my answer...

Answer 6

Add two constant functions, you have a constant function
Scalar multiple of a constant function, you have a constant function
Zero vector exists

Answer 7

A) fails scalar mult
B) ok
C) fails scalar mult
D) ok

Answer 8

what about e

Answer 9

Seems OK to me, passes all three tests.

Answer 10

D also fails addition property

Answer 11

D? you can add two constant functions to get a constant function

Answer 12

mistake i meant C fails addition property

Answer 13

is E is a subspace

Answer 14

is C fails addition property

Answer 15

yes because suppose you have two such functions, f + g (0) = 4

Answer 16

what about E

Answer 17

I think its a subspace, but verify yourself. Test all the axioms.

Answer 18

i think it is can you check please sir

Answer 19

k_1 + k_2*sin(x) + k_3 + k_4*sin(x) =
(k_1 + k3) + (k_2+ k_4)*sin(x)

closure under addition

zero vector where k_1= 0 and k_2=0

and scalar mult is fine

Answer 20

i do not understand zero vector can you please explain

Answer 21

A subspace must satisfy the following:
It must have a zero vector
It must be closed under addition
It must be closed under multiplication

Answer 22

If k_1 = 0 and k_2 = 0 then we have 0+ 0*sin(x) = 0