Projection of a vector onto direct sum subspace

Question

Define $\Pi(x|A)$ to be the unique projection of $x\in X$ onto the subspace $A$, for which $A\subset X$.

Is there a rigorous way of claiming that $\Pi(x|A\oplus B)=\Pi(x|A)$ for which: $B\subset X$, $x\perp B$, and $A\perp B$ (where $\perp$ means "orthogonal to")? It seems fairly intuitive that the subspace $B$ has no additional information about $x$ (by orthogonality), so that projecting $x$ onto the subspace of the direct sum $A\oplus B$ is equivalent to projection $x$ only onto $A$.

Answer 1

I assume that $X$ is a finite-dimensional inner product space. Let me write $p_A(x)$ for the projection of $x \in X$ onto the subspace $A$, although this carries the same information as your notation. Then you are correct that if $B$ is another subspace such that $A \perp B$ and $x \perp B$, then

$$p_{A \oplus B}(x) = p_A(x).$$

This is because of the following more general fact: if $A \perp B$, then

$$p_{A \oplus B}(x) = p_A(x) + p_B(x).$$

Now observe that if $x \perp B$, then $p_B(x) = 0$.