Question

Determine whether the following aare subspaces of C[-1,1]:

(a) The set of functions f in C[-1,1] such that f(-1)=f(1)

What is the procedure followed to proove these kind of questions ?

Answer 1

I assume you know the definition of a subspace?

Answer 2

yes

Answer 3

what does it tell you ... that you need to show?

Answer 4

closure under vector addition, scalar mult and that there is the 0 vector in it

Answer 5

ok

Answer 6

so if \(f\in C[-1,1]\) and \(f(-1)=f(-1)\) does it follow that

\[cf\in C[-1,1]\]

Answer 7

\(f(-1)=f(1)\)

Answer 8

ie is \(cf(-1)=cf(1)\)

Answer 9

i don't really understand why they need to point out that f(-1) = f(1)

Answer 10

they are just creating a subspace with certain properties.

Answer 11

if the axioms hold then it is a subspace

Answer 12

so is \(cf(-1)=cf(1)\)?

Answer 13

true

Answer 14

if \(f,g\in C[-1,1]\) does it hold that \(f+g\in C[-1,1]\)?

Answer 15

not necessarly

Answer 16

we should also note that if \(f,g\) are continuous then so are \(cf\) and \(f+g\)

Answer 17

why?

Answer 18

oh ok, i understand what you meant

Answer 19

if \(f,g\in C[-1,1]\) then

\(f(-1)=f(1)\) and \(g(-1)=g(1)\)

thus

\(f(-1)+g(-1)=f(1)+g(1)\)

Answer 20

ok?

Answer 21

I understand that

Answer 22

what is the zero vector?

Answer 23

f(0) ?

Answer 24

no...\(f(x):=0\)

Answer 25

then \(f\) is continuous and \(f(-1)=0=f(1)\)

thus \(f\in C[-1,1]\)

Answer 26

thank, for the help but who did u know that f(−1)=0=f(1) ?

Answer 27

if \(f\) is the zero vector...ie the zero function then \(f(x)=0\) for all \(x\). That includes -1 and 1