Linear Maps from Fⁿ to V are continuous

• categories: Functional analysis, Theorem

Every Linear Map from ${\mathbb{F}}^n$ to $V$ is Continuous

Let $\mathbb{F}$ denote either the field of real numbers $\mathbb{R}$ or complex numbers $\mathbb{C}$. Suppose $V$ is a normed vector space over $\mathbb{F}$. Then, every linear map $T:{\mathbb{F}}^n\to V$ is continuous.

Explanation and Proof Outline

1. Finite-Dimensional Domain: In the vector space ${\mathbb{F}}^n$, all norms are equivalent (see Equivalence of Norms in Fⁿ)
2. Linearity and Continuity: To show continuity, we only need to show that $T$ is bounded. Since $T$ is linear, continuity at a single point (e.g., $0$) implies continuity everywhere.
3. Boundedness of $T$: Given the equivalence of norms in ${\mathbb{F}}^n$, there exists a constant $C>0$ such that: $$\Vert T(x)\Vert \le C\Vert x\Vert$$ for all $x\in {\mathbb{F}}^n$. This inequality shows that $T$ is bounded with respect to any norm on ${\mathbb{F}}^n$ and hence is continuous.