Is this function continuous? (vector function) Assume you have $k$ vectors: $\{v_1,\dots,v_k\}$ in $\mathbb{R}^n$, and $\lambda\in\mathbb{R}^k$.
Look at the function:
$F\colon\mathbb{R}^k\rightarrow \mathbb{R}^n$
where $F(\lambda)=\Sigma_{i=1}^k\lambda_iv_i$, here $\lambda_i$ will be a scalar, and $v_i$will be a vector in $\mathbb{R}^n$.
My question is, will $F$ be continuous?, and does the answer depend on wheter the $v_i$-vectors are linearly independent?

Yes, $F$ is continuous. Let $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots,y_n)$ then
$$|F(x)-F(y)|\le \sum_{k=1}^n |x_k-y_k||v_k|\le |x-y|\sum_{k=1}^n |v_k|.$$
So the left side converges to 0 when $y\to x$. Continuity of $F$ does not depend on the linear independence of $\{v_i\}$.
