Evaluation \(x\mapsto x(v)\) is continuous by the product topology. Conversely, the inverse image of a basic box is the finite intersection \(\bigcap _v\{y:f(y)(v)\in O_v\}\), which is open if all coordinates are continuous. For a linear map \(T:X_V\to X_W\), the identity \((Tx)(w)=\sum _vx(v)(T\delta _v)(w)\) makes each output coordinate a finite linear combination of continuous inputs. Apply this also to the inverse when \(T\) is invertible. The empty product has the single basic box \(X_\varnothing \); the same statements hold directly.

Pointwise addition, scalar multiplication and \(h_V\) are continuous by finite sums and products of coordinates. The counting form is positive: if \(x\ne 0\), some \(x(v)^2>0\). Its perfectness can also be read directly: for a linear functional \(\ell \), put \(a(v)=\ell (\delta _v)\); then \(\ell (x)=h_V(a,x)\), uniquely by evaluation on the indicators. These are the real specializations of the cited field and duality constructions.