Question

State whether True or False.
Let 

C[0,1]

 be the space of all continuous real valued functions on 

[0,1]

.  For any continuous function 

K:[0,1]×[0,1]→R

, the map 

TK:C[0,1]→C[0,1]

 defined as

TK(f)(x)=∫10K(x,y)f(y)dy,x∈[0,1]

,
is a linear map.

True

 

False


​

Answer 1

State whether True or False.

Let  

C[0,1]

be the space of all continuous real valued functions on  

[0,1]

.  For any continuous function  

K:[0,1]×[0,1]→R

, the map  

TK:C[0,1]→C[0,1]

defined as

TK(f)(x)=∫10K(x,y)f(y)dy,x∈[0,1]

,

is a linear map.

True

 

False

Step-by-step explanation:

We first let the four consecutive terms in A.P. be= a-3d, a-d, a+d, a+3d

As per the first condition,

a-3d+a-d+a+d+a+3d=12

 4a=12

a= 12/4

a= 3 ......( eq.1)

As per the second condition,

a+d+a+3d=14

2a+4d= 14

2(3)+4d=14 (from eq.1)

6+4d=14

4d=14-6

 4d= 8

d=8/4

d=2

a-3d= 3-3(2)

         = 3-6= -3

a-d= 3-2= 1

a+d= 3+2=5

a+3d= 3+2(3)= 9

therefore The four consecutive terms of A.P. are -3, 1, 9 and 5.