Proving that something is a subspace of all the infinite sequences

  • #1

Homework Statement



let V be the vector space consisting of all infinite real sequences. Show that the subset W consisting of all such sequences with only finitely many non-0 entities is a subspace of V


The Attempt at a Solution


Ok so i have to show 1.Closure under Addition,2. Closure under Scalar Multiplcation

let x=(x[tex]_{n}[/tex]),y=(y[tex]_{n}[/tex]) [tex]\in[/tex] W
x + y [tex]\in[/tex] W closed under addition and since W is a set of finite sequences [tex]\in[/tex] V as it consists of infinite sequences

[tex]\lambda[/tex] is a scalar and [tex]\lambda[/tex]x [tex]\in[/tex] W closed under scalar multiplication and since W is a set of finite sequences [tex]\in[/tex] V as it consists of infinite sequences

I think this proves it

...But if [tex]\lambda[/tex] is negative this proof doesnt work,is there something i am missing or have i provided the required proof?
Thanks
 
Last edited:

Answers and Replies

  • #2
CompuChip
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You want to check your definitions for V and W.
W is not closed under addition because the sequences in W are finite and those in V are infinite... you actually told us yourself that V consists of finite sequences.
What you need to check is that when you add two sequences with finitely many non-zero entries, you get something with finitely many non-zero entries.

So either look carefully at your proof, or at your definitions, because it looks like the proof doesn't belong to the exercise.
 
  • #3


whoops really sorry V consists of all infinite real sequences.

if W only consists of finitely many entries and V consists of infinite does it not follow that W has to be a sub space?
 
  • #4
CompuChip
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Yes, it follows, but you still have to show it by checking the properties of a subspace.

For example: is W closed? I.e., if you take two infinite sequences with only finitely many non-zero entries, and you add them, do you get again a sequence with only finitely many non-zero entries?

What else do you have to show?
 
  • #5
HallsofIvy
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Suppose u has n non-zero entries and v has m non zero entries. What is the maximum possible number of non-zero entries of u+ v?
 

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