uniform convergence of a sequence of linear transformations on a compact subset

Question

Let $\{T_n\}$ be a sequence of linear transformations from $\mathbb{R}^k$ to $\mathbb{R}^m$. If $\{T_n\}$ converges pointwise to $T$, then

1. $T$ is a linear transformation;
2. the convergence is uniform on each compact subset of $\mathbb{R}^k$.

I'm especially confused about how to prove part 1.

Answer 1

Here is how you advance. Let $z=x+y \in \mathbb{R^k}, \forall x,y\in \mathbb{R^k} $, now by the linearity of $T_n$, we have

$$ T_n(z)=T_n(x+y) = T_n(x) + T_n(y), \quad \forall n\in \mathbb{ N} $$

$$ \implies \lim_{n\to \infty} T_n(z) = \lim_{n\to \infty} T_n(x)+ \lim_{n\to \infty} T_n(y)\implies T(z)=T(x+y) = T(x)+T(y),$$

by the pointwise convergence of $\left\{T_n \right\}$. Now, you need to prove the other condition for linearity

$$ T(ax)=aT(x). $$