I am new to VC dimension. I have to show that :
Let $g_1, \cdots, g_d$ : $\mathcal{X} \rightarrow \mathbb{R}$ arbitrary : show that :
$\mathcal{H} = \{ 1_{b_1 g_1(x) + \cdots + b_d g_d(x) \geq 0} , b_1, b_2 , \dots, b_d \in \mathbb{R} \}$ has VC dimension at most $d$.
I tried this but I am not sure. I take $d+1$ points $x_1, \cdots, x_{d+1}$ of $\mathcal{X}$. I note $y_i$ their labels
If I have at least two points $x_i$ and $x_j$ such that $b_1 g_1(x_i) + \cdots + b_d g_d(x_i) \geq 0$ and $b_1 g_1(x_j) + \cdots + b_d g_d(x_j) < 0$
Then I take $x_{d+1} = x_i$ and $y_{d+1} = 1_{b_1 g_1(x_j) + \cdots + b_d g_d(x_j) \geq 0}$
then $y_{d+1} \neq 1_{b_1 g_1(x_{d+1}) + \cdots + b_d g_d(x_{d+1}) \geq 0}$, and $y_{d+1}$ is missclassified.
If all of the points have the same label : ($1$ for example), I take $x_{d+1} = x_i$ ($i \in \{1,\dots,d\}$), and I take $y_{d+1} = 0$. So $x_{d+1}$ is missclassified.
Thanks in advance.