Matrix transformation and boundedness

Question

I have seem in books where they use the fact that a matrix transformation defined everywhere must be bounded. Can someone help me in understanding why this is true

Answer 1

The result is true in finite dimesnsionalspaces but false in infinite dimensional Hilbert spaces. It is well known that there exist discontinuous linear functionals in the latter case ( hence also discontinuous linear maps from the space into itself). So consider a finite dimensional sapce now. If $\{e_1,e_2,..,e_n\}$ is an orthogonal basis then any vector $v$ can be written as $\sum a_ie_i$. So we get $Av=\sum a_i Ae_i$. Let $C$ be the maximum of the numbers $\|Ae_i\|, 1 \leq i \leq n$. Then $\|Av\| \leq C\sum |a_i|\leq C\sqrt n(\sum |a_i|^{2})^{1/2} =C\sqrt n \|v\|$.