Linear Transformations between Finite-Dimensional Normed Vector Spaces are Continuous

Theorem
Linear transformations between finite-dimensional normed vector spaces are continuous.

Proof
We have that Norms on Finite-Dimensional Real Vector Space are Equivalent.

Choose the Euclidean norm.

Let $X = \struct {\R^n, \norm {\, \cdot \,}_2}$ and $Y = \struct {\R^m, \norm {\, \cdot \,}_2}$ be normed vector spaces.

Let the matrix $A \in \R^{m \times n}$ be given by:


 * $A = \begin {bmatrix}

a_{1 1} & \cdots & a_{1 n} \\ \vdots & \ddots & \vdots \\ a_{m 1} & \cdots & a_{m n} \\ \end{bmatrix}$

We have that Set of Linear Transformations is Isomorphic to Matrix Space.

Let $T_A : \R^n \to \R^m$ be the linear transformation such that:


 * $\forall \mathbf x \in \R^n : T_A \mathbf x := A \mathbf x$

Then:

Hence:


 * $\ds \norm{T_A \mathbf x}_2 \le \norm{\mathbf x}_2 \sqrt {\sum_{i \mathop = 1}^m \sum_{j \mathop = 1}^n a_{ij}^2 }$

Thus, $T_A$ is continuous.

Hence, all linear transformations between finite-dimensional normed vector spaces are continuous.