Showing any linear operator $T : X \to Y$ is bounded, where $X$ is a finite dimensional normed vector space, and $Y$ any normed vector space.

Question

Let $X$ be a finite dimensional normed vector space and $Y$ an arbitrary normed vector space. Show that any linear operator $T : X \to Y$ is bounded.

I got the hint to first show that $\| x\|_0 := \| x \| + \| Tx\|$, $x \in X$, defines a norm on $X$, but I do not know how this should help me.

Further I should calculate $\|T\|$ for where $X = K^n$, equipped with the Euclidean norm $\|\cdot\|_2$, $Y := \ell_1(\mathbb{N})$ and $Tx := (x_1,\ldots,x_n,0,0,\ldots) \in \ell_1(\mathbb{N})$, for all $x = (x_1,\ldots,x_n) \in K^n$.

Can please someone help?

Answer 1

Regarding the operator norm: I was thinking

$ ||T∥_2 = \sup \limits_{x \neq 0} \frac{∥Tx∥_1}{∥x∥_1} = \sup \limits_{x \neq0} \frac{∥( x_1,…,x_n,0,0,…)∥_1}{∥(x_1,…,x_n)∥_1} = \sup \limits_{x \neq0} \frac{|x_1|+…+|x_n|}{|x_1|+…+|x_n|}= 1 $