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Let $v_1$ and $v_2$ be two elements of a real Banach space $(X,\lVert\;\rVert)$, each of unit norm. Consider the one-dimensional affine subspace $v_1+tv_2$ for real $t$. Is there a general formula for the infimum $$\inf_{_t\in \mathbb{R}}||v_1+tv_2||$$ explicitly in terms of $v_1$ and $v_2$?

For example, in the case where $X$ is a Hilbert Space, it can be shown that the required formula is obtained when $t=-\langle\;v_1,v_2\;\rangle$.

I doubt there is such a formula. If the norm is differentiable, or can be made differentiable by composing it with some monotone differentiable function of $\mathbb{R}^{+}$, then perhaps there would be a chance at obtaining such a formula (simply by differentiating, setting that equal to zero, and trying to solve for the specific $t$).

If this is asking too much, it would also be nice to know if one can explicity write down a function $f(v_1,v_2)$ such that $$0 \leq f(v_1,v_2) \leq \inf_{_t\in \mathbb{R}}||v_1+tv_2||$$ and such that $f(v_1,v_2)=0$ only if $v_1$ and $v_2$ are parallel.

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More precisely I mean that $X$ is a real banach space, and $t$ ranges over the reals. I would like to know if one could explicitly find a real valued continuous function of $v_1$ and $v_2$ of absolute value strictly less than $||v_1 +tv_2||$ for all $t$. – student Oct 16 '13 at 15:08
Of course $\ f(v_1\ v_2) := -1\ $. Seriously, it's perhaps my own problem but I give up on many questions--I wish they would be summarized by a formal logical statement like $\ \forall_{a\ b}\exists_{c}\ \ldots\ $ and similar. – Włodzimierz Holsztyński Oct 16 '13 at 15:31
I'm sorry for being imprecise, but I meant a non-negative and (non-trivial) function. More strongly, is there a formula for the infimum over all $t$ of $||v_1+tv_2||$? – student Oct 17 '13 at 1:28
Thank you, @edger. You mean to discuss a generalization of cosinus from Hilbert spaces onto Banach spaces. One stage should be concerned with the 2-dimensional Banach spaces only. The second stage would address arbitrary dimension. That's how I see it. – Włodzimierz Holsztyński Oct 17 '13 at 15:46
This q. is about cosinus, sinus and a norm. Given $\ p\ v\in S\subseteq B$ of elements of norm $1$ one wonders about $\ ||a||_v\ :=\ \inf_{t\in\mathbb R} \sqrt{t^2+||p-t\cdot v||^2}\ $, etc. – Włodzimierz Holsztyński Oct 17 '13 at 16:16

Since $||v_1+tv_2||$ is a convex function of $t$, it is differentiable almosst everywhere and attains its minimum at the points where its derivative changes sign or is zero (the minimum does not have to be unique). If your data provides enough information to find such point, you are done. In general, you need to have a sufficient amount of data to say something about the problem.

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the most important fact to remember is probably that

for a linear subspace $S$ of a Banach space $X$, the projection operator : $$P_S(x) = \inf_{u \in S} \|x-u\|$$

is sublinear : (the result of the minimization is $u$, but for convenience assume the result can also be $\|x-u\|$ )

$$P_S(\alpha x) = \inf_{\alpha u \in S} \|\alpha x-\alpha u\| = |\alpha|\inf_{\alpha u \in S} \| x-u\| = |\alpha| P_S(x)$$

$$P_S(x+y) = \inf_{u,v \in S} \|x+y-(u+v)\| \le \inf_{u,v \in S} \|x-u\| + \|y-v\|$$ $$= \inf_{u \in S} \|x-u\|+\inf_{v \in S} \|y-v\| = P_S(x)+P_S(y)$$

I'm pretty sure this is why we require in the Hahn–Banach theorem that some linear map on a linear subspace $S$ of $X$ be bounded (on $S$) by some sublinear function on $X$.

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