Measure, Integration & Real Analysis, 2021a

224 Chapter 8 Hilbert Spaces
8B
Orthogonality
Orthogonal Projections
The previous section developed inner product spaces following a standard linear
algebra approach. Linear algebra focuses mainly on finite-dimensional vector spaces.
Many interesting results about infinite-dimensional inner product spaces require an
additional hypothesis, which we now introduce.
8.21 Definition Hilbert space
A Hilbert space is an inner product space that is a Banach space with the norm
determined by the inner product.
8.22 Example Hilbert spaces
• Suppose μ is a measure. Then L 2 (μ) with its usual inner product is a Hilbert
space (by 7.24).
• As a special case of the first bullet point, if n ∈ Z + then taking μ to be counting
measure on {1, . . . , n} shows that F n with its usual inner product is a Hilbert
space.
• As another special case of the first bullet point, taking μ to be counting measure
on Z + shows that l 2 with its usual inner product is a Hilbert space.
• Every closed subspace of a Hilbert space is a Hilbert space [by 6.16(b)].
8.23 Example not Hilbert spaces
• The inner product space l 1 , where 〈(a 1 , a 2 ,...), (b 1 , b 2 ,...)〉 = ∑ ∞ k=1 a kb k ,is
not a Hilbert space because the associated norm is not complete on l 1 .
• The inner product space C([0, 1]) of continuous F-valued functions on the interval
[0, 1], where 〈 f , g〉 = ∫ 1
0
f g, is not a Hilbert space because the associated
norm is not complete on C([0, 1]).
The next definition makes sense in the context of normed vector spaces.
8.24 Definition distance from a point to a set
Suppose U is a nonempty subset of a normed vector space V and f ∈ V. The
distance from f to U, denoted distance( f , U), is defined by
distance( f , U) =inf{‖ f − g‖ : g ∈ U}.
Notice that distance( f , U) =0 if and only if f ∈ U.
Measure, Integration & Real Analysis, by Sheldon Axler