Measure, Integration & Real Analysis, 2021a

Section 8B Orthogonality 231
The next result states that if U is a
closed subspace of a Hilbert space V,
then V is the direct sum of U and U ⊥ ,
often written V = U ⊕ U ⊥ , although
we do not need to use this terminology
or notation further.
The key point to keep in mind is
that the next result shows that the picture
here represents what happens in
general for a closed subspace U of a
Hilbert space V: every element of V
can be uniquely written as an element
of U plus an element of U ⊥ .
8.43 orthogonal decomposition
Suppose U is a closed subspace of a Hilbert space V. Then every element f ∈ V
can be uniquely written in the form
f = g + h,
where g ∈ U and h ∈ U ⊥ . Furthermore, g = P U f and h = f − P U f .
Proof
Suppose f ∈ V. Then
f = P U f +(f − P U f ),
where P U f ∈ U [by definition of P U f as the element of U that is closest to f ] and
f − P U f ∈ U ⊥ [by 8.37(a)]. Thus we have the desired decomposition of f as the
sum of an element of U and an element of U ⊥ .
To prove the uniqueness of this decomposition, suppose
f = g 1 + h 1 = g 2 + h 2 ,
where g 1 , g 2 ∈ U and h 1 , h 2 ∈ U ⊥ . Then g 1 − g 2 = h 2 − h 1 ∈ U ∩ U ⊥ , which
implies that g 1 = g 2 and h 1 = h 2 , as desired.
In the next definition, the function I depends upon the vector space V. Thus a
notation such as I V might be more precise. However, the domain of I should always
be clear from the context.
8.44 Definition identity map; I
Suppose V is a vector space. The identity map I is the linear map from V to V
defined by If = f for f ∈ V.
The next result highlights the close relationship between orthogonal projections
and orthogonal complements.
Measure, Integration & Real Analysis, by Sheldon Axler