Q:

Show that if $\dim(E)=n < \infty$, then there exists an operator $T:E \to \ell_p$ such that $\lambda^{-1} \| x \| \leq \| Tx \| \leq \| x \|$

Suppose $X$ is a Banach space and $E$ is a subspace of $X$.
Show that if $\dim(E)=n < \infty$, then there exists an operator $T:E \rightarrow \ell_p$ such that $$\lambda^{-1} \| x \| \leq \|  Tx  \| \leq \| x \|$$
for some $\lambda \in \mathbb{R}$.

Since $E$ is finite dimensional, any norm on it is equivalent. But from here onwards, I have no idea how to proceed.

A:

Choose a basis $e_1,\ldots,e_n$ of $E$ and define $T$ on the basis (and then by linear extension) by $T(e_k)=(\delta_{n,k})_{n\in\mathbb N}$.

