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

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!!

-
What have you tried? –  Matthew Pressland Mar 6 '13 at 11:03
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

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.

-
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
@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