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I'm going through some PDE notes and found a statement without proof, name or reference and was wandering whether anybody had any of these:

Let $X,Y$ be Banach spaces and let $E$ be a dense linear subspace of X. If T is continous from $E$ to $Y$ (i.e. $\exists c>0$ s.t. $||Te||_Y\leq c ||e||_X$ for all $e\in E$). Then T has a unique extension $\bar{T}:X\rightarrow Y$ with $\bar{T}|_E=T$ satisfying $||Tx||_Y\leq c ||x||_X$ for all $x\in X$.

I was also wondering if it was ommited whether or not $T$ respectively $\bar{T}$ are linear or whether the statement doesn't hold in the case of linearity (I'm aware that continuity and boundedness are equivalent in the case of linear operators, but I found it unusual that a reference to a continous operator is being made here using a definition of continuity that more closely resembles boundedness).

Goofy
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  • It's about linear operators. It holds for general uniformly continuous functions as well https://math.stackexchange.com/questions/76854/extending-a-function-by-continuity-from-a-dense-subset-of-a-space – Evangelopoulos Foivos Feb 19 '25 at 20:34

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