I find it helpful to break down the Galois connection requirement into the "if" and the "only if". Eg for \\(f(a) = 2a\\), I label the left and right adjoints \\(L\\) and \\(R\\), respectively, and look at the various conditions \\(\\{C_{implication}\\}\\):


(least) $$C_{f \leftarrow R}: f(a) \leq b \Leftarrow a \leq R(b) \textrm{ means } R(b) \leq_{\Bbb{R}} \Bigl\lfloor \frac{b}{2} \Bigr\rfloor$$

(upper bound) $$C_{f \to R}: f(a) \leq b \Rightarrow a \leq R(b) \textrm{ means } R(b) \geq_{\Bbb{R}} \Bigl\lfloor \frac{b}{2} \Bigr\rfloor$$

Similarly,

(greatest) $$C_{L \to f}: L(b) \leq a \Rightarrow b \leq f(a) \textrm{ means } L(b) \geq_{\Bbb{R}} \Bigl\lceil \frac{b}{2} \Bigr\rceil$$

(lower bound) $$C_{L \leftarrow f}: L(b) \leq a \Leftarrow b \leq f(a) \textrm{ means } L(b) \leq_{\Bbb{R}} \Bigl\lceil \frac{b}{2} \Bigr\rceil$$



In the vector subspace linear transform example,

$$C_{T \leftarrow R}: T(v) \subseteq w \Leftarrow v \subseteq R(w)$$ means \\(R(w) = R(w \cap Image(T)) \; \forall w \subseteq W\\), i.e. \\(Image(T)^{\perp}\\) is in a very loose sense in the "kernel" of \\(R\\) (\\(R(Image(T)^{\perp}) = R(\Bbb{0})\\)), and \\(R\\) "at most" gives you the linear inverse on \\(Image(T)\\),

$$C_{T \to R}: T(v) \subseteq w \Rightarrow v \subseteq R(w)$$ means \\(Ker(T) \subseteq R(w) \; \forall w \subseteq W\\) and \\(R\\) "at least" gives you the linear inverse on \\(Image(T)\\),

$$C_{L \to T}: L(w) \subseteq v \Rightarrow w \subseteq T(v)$$ means \\(w \subseteq Image(T) \; \forall w \subseteq W\\), i.e. \\(Image(T)^{\perp} = \Bbb{0}\\) (otherwise \\(L\\) is not defined), and \\(L\\) is "at least" the linear inverse on \\(Image(T)\\), and

$$C_{L \leftarrow T}: L(w) \subseteq v \Leftarrow w \subseteq T(v)$$ means \\(L(w) \cap Ker(T) = \Bbb{0} \; \forall w \subseteq W\\) and \\(L\\) is "at most" the linear inverse on \\(Image(T)\\).

\\(C_{T \leftarrow R}\\) gives you "least", \\(C_{T \to R}\\) gives you "upper bound"; together, \\(R\\) is the "least upper bound"; \\(C_{L \to T}\\) and \\(C_{L \leftarrow T}\\) are "greatest", "lower bound", respectively.