Extension of bounded operators between norm spaces

Question

Let $X$ and $Y$ be two Banach Spaces and $X_1$ be a subspace of $X$. If $T$ is a bounded linear operator from $X_1\to Y$, then,

1. this is an extension of $T$ from $X\to Y$ such that $\|T\|_{X}\le \|T\|_{X_1}$.

2. The extension is unique if $X_1$ is dense in $X$.

Could some please help with the proof? Or provide a reference where I can find an answer. I have been flipping through all my text and urgently need something quick when studying the chapter of Sobolev space.(There are quite a few extension theorem I've seen in books. However every version is slightly different. Hence I hope someone can guide me through as an example.)

Answer 1

The following claim is not true in general.

Let $X$ and $Y$ be Banach spaces and $X_1 \subset X$ a subspace. If $T: X_1 \to Y$ is a continuous bounded operator, then there exists a continuous linear extension to the whole space $X$, i.e. there exists a bounded linear operator $\hat T: X \to Y$ such that $ \hat{T}\restriction_{X_1} = T$.

Take for example $X=\ell^\infty$ $X_1 = c_0$ and $Y=c_0$. Now, the identity $\text{id}: c_0\to c_0$ is clearly a bounded linear operator but has no bounded extension to $\ell^\infty$. If we had one, this would be a continuous decomposition of the form $\ell^\infty = c_0 \oplus V$. This cannot happen due to Phillip's lemma, see this MSE thread.

However, if $X_1\subset X$ is a dense subspace then there is a unique continuous extension. This is essentially the continuous linear extension theorem. In general, there exist also unbounded extensions, however, only one bounded extension.