Linear maps between finite dimensional spaces

Is it true that any linear map between two arbitrary finite-dimensional vector spaces is continuous? Is it differentiable?

Is it true that any linear map between two arbitrary finite-dimensional vector spaces is continuous? Is it differentiable?

This is indeed true! Take an arbitrary linear map $f:\mathbb{R}^n\rightarrow \rightarrow{R}^m$. Then it suffices that the components f1,...,fm are continuous. And this is true since

$$\begin{eqnarray*} |f_1(x_1,...,x_n)| & = & |x_1f_i(1,0,...,0)+...+x_nf_i(0,...,0,1))|\\ & \leq & |x_1||f(1,0,...,0)|+...+|x_n||f(0,...,0,1)|\\ & \leq & \|(x_1,...,x_n)\|_\infty(|f(1,0,...,0)|+...+|f(0,...,0,1)|)\\ \end{eqnarray*}$$

which implies continuity. For differentiability, we have to find a linear map u such that

$$\lim_{h\rightarrow 0}{\frac{f(x_0+h)-f(x_0)-u(h)}{\|h\|}}=0$$

take u=f and it's easy to see that this is true.

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Thanks! Also, if a map is differentiable, it is also continuous and so the proof can be made really simple, right?

BTW, could you explain to me how to type latex code?

Thanks! Also, if a map is differentiable, it is also continuous and so the proof can be made really simple, right?

Yes, but the proof that a differentiable map is continuous uses that linear maps are continuous... (at least the proof I've seen)

BTW, could you explain to me how to type latex code?

Click on the LaTeX symbols to see what I did.

mathwonk