Equivalent conditions for a linear map between two normed spaces to be continuous everywhere
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Statement
Given two normed spaces [ilmath](X,\Vert\cdot\Vert_X)[/ilmath] and [ilmath](Y,\Vert\cdot\Vert_Y)[/ilmath] and a linear map [ilmath]L:X\rightarrow Y[/ilmath] between them, then[1] the following are equivalent (meaning if you have 1 you have all the others):
- If we have a sequence [ilmath](x_n)_{n=1}^\infty\subseteq X[/ilmath] with [ilmath]x_n\rightarrow 0[/ilmath] then [ilmath](\Vert L(x_n)\Vert_Y)_{n=1}^\infty[/ilmath] is a bounded sequence
- [ilmath]L[/ilmath] is continuous at a point (any point)
- [ilmath]L[/ilmath] is a bounded linear map, that is [ilmath]\exists A>0\ \forall x\in X[\Vert L(x)\Vert_Y\le A\Vert x\Vert_X][/ilmath]
- [ilmath]L[/ilmath] is continuous everywhere
Proof
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See page 154 in[1]
References
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