3 KiB
Direct sums
Definition 1: in a metric space
(X,d)
, the distance\delta
from an elementx \in X
to a nonempty subsetM \subset X
is defined as
\delta = \inf_{\tilde y \in M} d(x,\tilde y).
In a normed space (X, \|\cdot\|)
this becomes
\delta = \inf_{\tilde y \in M} |x - \tilde y|.
Definition 2: let
X
be a vector space and letx, y \in X
, the line segmentl
between the vectorsx
andy
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 spaceX
is convex if for allx, y \in M
the line segment betweenx
andy
is contained inM
.
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 letM \subset X
be a complete convex subset ofX
. Then for everyx \in X
there exists a uniquey \in M
such that
\delta = \inf_{\tilde y \in M} |x - \tilde y| = |x - y|,
if
M
is a complete subspaceY
ofX
, thenx - y
is orthogonal toX
.
??? note "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 sumX = Y \oplus Z
of two subspacesY \subset X
andZ \subset X
ofX
if eachx \in X
has a unique representation
x = y + z,
for
y \in Y
andz \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 spaceX
, then
X = Y \oplus Y^\perp,
with
Y^\perp = \{x\in X \;|\; x \perp Y\}
the orthogonal complement ofY
.
??? note "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 spaceX
and letP: 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\}
.
??? note "Proof:"
Will be added later.
Lemma 2: if
Y
is a closed subspace of a Hilbert spaceX
, thenY = Y^{\perp \perp}
.
??? note "Proof:"
Will be added later.
Then it follows that X = Y^\perp \oplus Y^{\perp \perp}
.
??? note "Proof:"
Will be added later.
Lemma 3: for every non-empty subset
M \subset X
of a Hilbert spaceX
we have
\mathrm{span}(M) \text{ is dense in } X \iff M^\perp = {0}.
??? note "Proof:"
Will be added later.