A question about usual inner product

Question

Let $H$ be an $m$-dimensional real vector space of functions on a set $X$, let $f$ be any real function on $X$, and let $H_1 := \{f + h : h \in H\}$.

I had these two questions below if anybody could help me out!

1. Why we can say surely $card(X)\geq m$?

2. Let G be the vector space $\{af + h : a \in \mathbb{R}, h \in H\}$. Let $r_A : G \mapsto \mathbb{R}^A$ be the restriction of functions in $G$ to $A$. If $r_A$ is not onto, take $0 \neq v \in \mathbb{R}^A$ where $v$ is orthogonal to $r_A(G)$ for the usual inner product $(.,.)_A$.
   What is this inner product exactly? and why there is such a vector $v$?

Answer 1

For 1: Hint: $H$ has a basis. How big is the basis? If $card X = k<m$, what's the biggest basis you can construct for a space of functions on $X$?