TITLE: How can this be a vector space?
QUESTION [5 upvotes]: I found the following statement:
"Example of a linear - vector - space:
The set $C^{(k)}[a,b]$ of all (real-valued) continuous functions on a finite
interval $a ≤ t ≤ b$ with addition and real number multiplication
1) $( f + g)(t) = f (t)+ g(t )$
2) $(α f )(t )= \alpha f (t )$, $t∈[a,b]$
forms a linear space. The zero vector $Θ: f (t )= 0$ for all $t∈[a,b]$."
I am a bit confused with 2). 
Take $f(x) = x^2$. Let $\alpha = 2$, then:
$2*x^2 = 2*f(2) = 2*4 = 8  \neq  2^2*x^2 = f(2*2) = 4*4 = 16 $
Hence $(α f )(t ) \neq \alpha f (t )$
So how is it that degree two polynomials form a linear space?
Added: I realize I had a typo and that my question makes little sense. Thanks Race Bannon for pointing this out.

REPLY [11 votes]: You calculated $f(\alpha x)$ instead of $(\alpha f)(x)$ in your example.