Confused about a part of the proof for $\mathcal{L}(V, W)$ is Banach provided that $V$ is normed and $W$ is Banach.

Question

Let $\mathcal{L}(V, W)$ be the set of linear maps where $V$ is normed and $W$ is Banach. In Analysis Now, Pedersen states that $\{T_n\} \subset \mathcal{L}(V,W)$ is Cauchy, then $\{T_n(x)\} \subset W$ is Cauchy in $W$. Even though this makes intuitive sense, I was trying to show this. Note, I am using the operator norm. Chatgbt says that this is true $$||(T_n-T_m)x|| \leq ||T_n - T_m||||x||$$ note that I am taking different norms, but I can't seem to prove it. Sorry if this is obvious like Pedersen presents it, and my linear algebra is definitely rusty.

Answer 1

By definition, $$\Vert T \Vert = \sup\{\Vert Tx \Vert : x \in X, \Vert x \Vert \leq 1\}.$$ For any non-zero $x \in X$, using the homogeneity of $T$ we get $$\Vert Tx \Vert = \left\Vert \Vert x \Vert T\left(\frac{1}{\Vert x \Vert}x\right)\right\Vert \leq \Vert x\Vert \Vert T \Vert = \Vert T \Vert \Vert x \Vert.$$

Now apply this in your case with $T = T_n - T_m$.