Direct sums
Definition 1: in a metric space \((X,d)\), the distance \(\delta\) from an element \(x \in X\) to a nonempty subset \(M \subset X\) is defined as
\[ \delta = \inf_{\tilde y \in M} d(x,\tilde y). \]
In a normed space \((X, \|\cdot\|)\) this becomes
Definition 2: let \(X\) be a vector space and let \(x, y \in X\), the line segment \(l\) between the vectors \(x\) and \(y\) is defined as
\[ l = \{z \in X \;|\; \exists \alpha \in [0,1]: z = \alpha x + (1 - \alpha) y\}. \]
Using definition 2, we may define the following.
Definition 3: a subset \(M \subset X\) of a vector space \(X\) is convex if for all \(x, y \in M\) the line segment between \(x\) and \(y\) is contained in \(M\).
This definition is true for projections of convex lenses which have been discussed in optics.
We can now provide the main theorem in this section.
Theorem 1: let \(X\) be an inner product space and let \(M \subset X\) be a complete convex subset of \(X\). Then for every \(x \in X\) there exists a unique \(y \in M\) such that
\[ \delta = \inf_{\tilde y \in M} \|x - \tilde y\| = \|x - y\|, \]if \(M\) is a complete subspace \(Y\) of \(X\), then \(x - y\) is orthogonal to \(X\).
Proof:
Will be added later.
Now that the foundation is set, we may introduce direct sums.
Definition 4: a vector space \(X\) is a direct sum \(X = Y \oplus Z\) of two subspaces \(Y \subset X\) and \(Z \subset X\) of \(X\) if each \(x \in X\) has a unique representation
\[ x = y + z, \]for \(y \in Y\) and \(z \in Z\).
Then \(Z\) is called an algebraic complement of \(Y\) in \(X\) and vice versa, and \(Y\), \(Z\) is called a complementary pair of subspaces in \(X\).
In the case \(Z = \{z \in X \;|\; z \perp Y\}\) we have that \(Z\) is the orthogonal complement or annihilator of \(Y\). Also denoted as \(Y^\perp\).
Proposition 1: let \(Y \subset X\) be any closed subspace of a Hilbert space \(X\), then
\[ X = Y \oplus Y^\perp, \]with \(Y^\perp = \{x\in X \;|\; x \perp Y\}\) the orthogonal complement of \(Y\).
Proof:
Will be added later.
We have that \(y \in Y\) for \(x = y + z\) is called the orthogonal projection of \(x\) on \(Y\). Which defines an operator \(P: X \to Y: x \mapsto Px \overset{\mathrm{def}}= y\).
Lemma 1: let \(Y \subset X\) be a subset of a Hilbert space \(X\) and let \(P: X \to Y\) be the orthogonal projection operator, then we have
- \(P\) is a bounded linear operator,
- \(\|P\| = 1\),
- \(\mathscr{N}(P) = \{x \in X \;|\; Px = 0\}\).
Proof:
Will be added later.
Lemma 2: if \(Y\) is a closed subspace of a Hilbert space \(X\), then \(Y = Y^{\perp \perp}\).
Proof:
Will be added later.
Then it follows that \(X = Y^\perp \oplus Y^{\perp \perp}\).
Proof:
Will be added later.
Lemma 3: for every non-empty subset \(M \subset X\) of a Hilbert space \(X\) we have
\[ \mathrm{span}(M) \text{ is dense in } X \iff M^\perp = \{0\}. \]
Proof:
Will be added later.