Continuous Family of Norms&Operators

Question

I want to ask the definition of a concept I see around but has not been able to find the definition for. Given a finite dimensional vector bundle $\pi : E \rightarrow K$ over a normed convex vector space K (with isomorphic fibers), what would be the definition of a continuous family of norms {$|.|_{x\in K}$}? I have two guesses but I don't know which one is correct, or if any of it is correct,

1- It is continuous as a map, $|.|: E \rightarrow R$ in the sense that for any sequence $v_n \rightarrow v$ in $E$ with $\pi(v_n)=x_n$, we have $|(v_n)_{x_n}|_{x_n}\rightarrow |(v)_{x}|_{x}$.

2- Or, first we choose a basis for each fiber $E_x$ write an isomorphism $T_{xy} :E_x \rightarrow E_y$, between each fibers. And then given a vector $v_x$ in the fiber $E_x$ and a sequence of points $x_k$ in K converging to x, $|T_{x_nx}(v_x)|_{x_n} \rightarrow |v_x|_{x} $

The difference between the two is whether if you fix vector component $v_x$ or now. I could ask the same question about continuous family of operators $T_x :E \rightarrow E$, is it continuity as a map from E to E or should we somehow again identify each fiber like in the second case above.

Thanks

Answer 1

One would be far more likely to define such a notion with something like your first definition, though the property people would really care about is that for any continuous section $\xi$ of your vector bundle $E \to X$, $x \mapsto |\xi(x)|$ defines a continuous map $X \to \mathbb{R}$. In particular, for continuity to make sense, all you need is for $X$ to be a topological space.

As for what actually tends to appear in practice, one usually doesn't just want a continuous family of norms but rather a metric, a continuous family $g$ of inner products in the sense that for any continuous sections $\xi$ and $\eta$ of $E \to X$, $x \mapsto g(x)(\xi(x),\eta(x))$ defines a continuous map $X \to \mathbb{R}$ (or $\mathbb{C}$, if $E \to X$ is complex). Again, you simply need $X$ to be a topological space. An example, of course, is the Riemannian metric on a Riemannian manifold $X$, which is a metric on the tangent bundle $TX \to X$, though more generally, in the case of smooth vector bundles over smooth manifolds, one usually wants metrics to be smooth in the same sense.