Let $X$ be a Banach space and let $W$ be a linear subspace. Consider some $v$ in $X$ and let the projection of $v$ onto $W$ be defined as the $u$ in $W$ st. $\forall w\in W,\|v-u-w\|=\|v-u+w\|$. If there is a unique $u$ for which this holds, call this $u$ the projection of $v$ onto $W$.
I would like to know if this notion of projection is well defined, that is, does $u$ exist and is $u$ unique.
For general Hilbert spaces $H$ and the usual definition of the orthogonal projection $P:H \rightarrow H$ onto a closed subspace $W \subset H$, we have $P(v) \in W$ satisfying $$ ||v - P(v) - w|| = ||v - P(v)|| + ||w|| = ||v - P(v) + w||, \forall w \in W $$ where an orthogonal decomposition was used (Parseval's identity for Hilbert spaces). Actually, using a parallelogram rule like expansion we can see $$ ||(v - u) - w||^2 - ||(v-u) + w||^2 = 0 \quad \iff \quad \langle (v-u),w \rangle = 0, \forall w \in W. $$ From this it follows that $(v-u) \in W^\perp$, and that your definition is equivalent to the the usual definition of the orthogonal projection on Hilbert spaces, unique since $v$ has a unique expansion with respect to $W \oplus W^\perp$.
This also means that for a general non-closed subspace of a Hilbert space, the usual definition of orthogonal complement is not well-defined (i.e. does not exist, I do not think uniqueness is ever a problem due to strict convexity of the norm, but you can ignore this comment), so your definition isn't as well.
For general Banach spaces I am not sure to be honest, maybe someone else can shed some light on this.
The usual orthogonal projection for Hilbert spaces can be generalised to Banach spaces in the following way: As you probably know, another possible way (as opposed to the inner product way) is to project to the unique $u \in W$ such that $||v - u||$ is minimized. Of course only a norm is used here, so it can be generalised to Banach spaces. However, it seems that for general subsaces it is not certain that this minimum is attained, as can be read on p. 133 of Conway, an introduction to functional analysis (http://users.math.uoc.gr/~frantzikinakis/FunctionalGrad2015/Conway.pdf). The generalisation goes well at least for reflexive Banach spaces (again p.133): There Conway states:
"It is not generally true that the distance from a point to a linear subspace is attained. If $M \subset X$ [$M$ is a vector subspace of a Banach space $X$] call $M$ proximinal if for every $x \in X$ there is a $y \in M$ such that $|| x-y || = dist(x, M)$. So if $X$ is reflexive, Corollary 4.6 implies that every closed linear subspace of $X$ is proximinal". (Compare this, if you have seem some non-linear functional analysis/calculus of variations, to the fact that the above norm, for a reflexive separable Banach space $X$, has a unique minimum, since the norm is weakly sequentially lower semicontinuous and weakly sequentially precompact and strictly convex.)
Again, I cannot see immediately how your definition relates to the one of proximinal subspaces and the generalisation of the orthogonal projection in Banach spaces. The only thing I can say, together with the Calculus of Variations perspective taken above is that, if the function $f:\mathbb{R} \rightarrow \mathbb{R}$ (or $\mathbb{C}$), $f_w(\lambda) = ||v -u - \lambda w|| - ||v - u + \lambda w||$ is differentiable at zero, for all $w \in W$, then I do believe find that the two notions/generalisations are equavalent.
