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Prove-that $W = \{f \mid f(230) = 0\}$ is a subspace of $F(-\infty,\infty)$.

I know that it has to be closed under addition and scalar multiplication, but i have no idea where to start. Please help asap!!

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What have you tried? –  Matt Pressland Mar 6 '13 at 11:03
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As I mentioned on your earlier question: To get the best possible answers, you should explain what your thoughts on the problem are so far. That way, people won't tell you things you already know, and they can write answers at an appropriate level; also, people tend to be more willing to help you if you show that you've tried the problem yourself. If this is homework, please add the [homework] tag. –  Zev Chonoles Mar 6 '13 at 11:04
    
i dont know where to start –  nicholas Mar 6 '13 at 11:28
    
i dont even know what the function of "f" is so where would i begin? –  nicholas Mar 6 '13 at 11:30
    
There is no one "function f"... –  vonbrand Mar 6 '13 at 14:48

1 Answer 1

You need to answer three questions. Assuming the operations on $\,V:=F(-\infty\,,\,\infty)\,$=the real vector space of (continuous, derivable or just...) functions defined on the whole real line, are defined to be

$$\forall\,f,g\in V\,,\,\forall\,k\in\Bbb R: (f+g)(x):=f(x)+g(x)\;,\;(kf)(x):=k(f(x))\;\,,\;x\in\Bbb R$$

then you have to answer the following questions

(1) Is the zero function in $\,W\,?\,$ , i.e.:$\;0(230)\stackrel ?=0\,$

(2) Is the sum of two elements of $\,W\,$ again an element of $\,W\,$ ? , i.e. : $$ f\,,\,g\in W\,\stackrel ?\Longrightarrow f+g\in W\,?$$

or what is the same: $$f,g\in W\stackrel ?\Longrightarrow (f+g)(230)=0$$

3) Finally, is the multiplication of an element in $\,W\,$ again an element of $\,W\,$ ?:

$$ f\in W\;,\;k\in\Bbb R\stackrel ?\Longrightarrow (kf)(230)=0$$

Even a mildly good understanding of the above makes the exercise utterly trivial, so be sure to understand the definitions and things you're working with.

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what does the upside down A mean, i forgot lol –  nicholas Mar 6 '13 at 11:34
    
nevermind its "for all" –  nicholas Mar 6 '13 at 11:40
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@nicholas , to avoid getting answers with stuff you don't understand you should make clear what your mathematics level is. –  DonAntonio Mar 6 '13 at 11:43

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