Show that the following subsets Ware vector subspaces of V.

Question

W c ([0,1]) and = Here is V = F([0,1] IR) Vector space of all function.+



F (0,1) - IR under operation.



(f+9) (a) f (a) + 9(a) = E [0,1] (xf) (a) αf(a) + & E [0,1] aEIR =



W set of all continuous function F (0, 1) TR =



The zero vector (ō) v is the zero function 0 a € [0,1] = ō (2)



We know that every constant function is continuous.



ō (a) is continuous function on [0,1]



DE W



W + Φ



Let α В e IR f g E W



i.ef and g are continuous function.



We know that scalar multiplication of continuous function is conti- nuous.



Also sum of continuous fun" is continuous



+ Bg is continuous

Answer 1

W c ([0,1]) and = Here is V = F([0,1] IR) Vector space of all function.+

F (0,1) - IR under operation.

(f+9) (a) f (a) + 9(a) = E [0,1] (xf) (a) αf(a) + & E [0,1] aEIR =

W set of all continuous function F (0, 1) TR =

The zero vector (ō) v is the zero function 0 a € [0,1] = ō (2)

We know that every constant function is continuous.

ō (a) is continuous function on [0,1]

DE W

W + Φ

Let α В e IR f g E W

i.ef and g are continuous function.

We know that scalar multiplication of continuous function is conti- nuous.

Also sum of continuous fun" is continuous

+ Bg is continuous