Question

my test is tomorrow can someone can me out
1)V ={f|f:N->R,lim(n->infinity)f(n)∈R}⊂R
Is this a vector space? Explain why or why not.
2)V ={f|f:N->R,lim(n->infinity)f(n)doent exist or(+-)infinity}⊂R^N
Is this a vector space? Explain why or why not.

Answer 1

answer for 1)for the first one i think it is yes cause as the limit approaches 0 the sum of it also and multiple the sequnce that approaches 0 is also approaches 0 ?

Answer 2

1) yes, it's a vector space.
f + g must be in that space
cf must be also in that space.

since f and g are convergent f+g is also convergent for all f and g
cf is also convergent

Answer 3

c is a scalar btw

Answer 4

thanks is the second one no?

Answer 5

second one is little bit tough :/

Answer 6

Let g = -f

Answer 7

Alternatively, let c = 0

Answer 8

@eva12 did you meant to type R^N for the second question? or just R?

Answer 9

R^N meaning (real numbers)^(natural numbers) for second one

Answer 10

ok? what what does R^N mean?

Answer 11

you mean like R^3, R^4,... ??

Answer 12

ch5 hears the book i am using

Answer 13

in the book its question #3 in ch5

Answer 14

oh i see now. R^N just means a set of all functions that map the naturals to the reals. I thought R^N is like a vector in R^1, R^2, r^3.. :D

Answer 15

Second one is not a vector space. As Wio said, it failed in the case c = 0
because 0 * lim f(n) = 0, which is not in the space
n->inf