Why stronger norm defines weak local minimizer?

Question

Why the stronger norm defines weak local minimizer, while the weaker norm defines strong local minimizer?

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For example, when minimizing a functional on $C^1[a,b]$, one can also consider the weaker norm coming from $C[a,b]$ (uniform norm), and define local minimizers using that norm.

Answer 1

A local minimizer beats all competitors in some neighborhood. In a stronger norm, neighborhoods are smaller, and therefore a local minimum is easier to have. Hence, being a local minimizer in a stronger norm is a weaker property than being a local minimizer in a weaker norm.

Such terminological switches happen all the time. For example, if topology $\mathcal T_1$ is weaker than topology $\mathcal T_2$ on the same set $X$, then

1. A set being open in $\mathcal T_1$ is a stronger property than being open in in $\mathcal T_2$
2. A set being compact in $\mathcal T_1$ is a weaker property than being compact in $\mathcal T_2$
3. A function being continuous on $(X,\mathcal T_1)$ is a stronger property than being continuous on $(X,\mathcal T_2)$.
4. For a function from $\mathbb R$ to $(X,\mathcal T_1)$, being continuous is a weaker property than being continuous to to $(X,\mathcal T_1)$.
5. and so forth...