Download Abstract Euclidean Space and l2

Intuitionism and Hilbert Space
University of Victoria, 2017
Abstract
In the proceeding paper we will slowly uncover an intuitionistically acceptable generalization
of Euclidean space whose classical analog is used extensively in analysis. In particular, we will
constructively define the mathematical object `2 pNq, and show that our construction satisfies
the definitive properties of our generalization. Many doubted that mathematical analysis had a
place in Brouwer’s original conception of intuitionistic mathematics. However, since a direct
analog of real numbers was found for the intuitionist, as well as a concept of measure [1], much
of this doubt has thankfully fallen away. The intent of the following discussion is to further their
inclusion of the intuitionist analyst into the mathematical community.
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Euclidean Space and `2 pNq
Classically speaking, the n-dimensional Euclidean space is the set of n-tuples of complex
numbers, Cn . This section will mostly be a description of the generalizaton of these spaces to
infinite dimensions, `2 pNq.
The mathematical object `2 pNq is defined classically to be the set of sequences of squaresummable complex numbers. In other words, if px1 , x2 , x3 , ¨ ¨ ¨ q is a sequence in C, px1 , x2 , ¨ ¨ ¨ q
belongs to `2 pNq if and only if there is a natural number M such that for any n in Z` the quantity
|x1 |2 ` |x2 |2 ` ¨ ¨ ¨ ` |xn |2 is bounded above by M . The square-summable condition on the
coordinates of a point in `2 pNq, though it may seem mysterious, will help us to leverage the
properties of everyday Euclidean space to the object `2 pNq: Consider the vector x in Euclidean
space with the coordinates x1 , ¨ ¨ ¨ , xn . The Euclidean distance from the origin to x is the
quantity
g
f n
fÿ
}x}n “ e
|xk |2 ,
k“1
as per Pythagoras’ theorem. The properties of this quantity important to any analyst, classical
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or intuitionist, are
1. if }x}n “ 0, then x “ 0,
2. (the Triangle Inequality) }x ` y}n ­ą }x}n ` }y},
3. (the Parallelogram Law) }x ` y}2n ` }x ´ y}2n “ 2}x}2n ` 2}y}2n .
(1)
4. Every sequence of points in Euclidean space which is Cauchy (see
[5]) with respect to } ¨ } converges to a point in Euclidean space.
The first property states that the only point which is without distance from the origin must
be the origin itself. The second property states that the length of the hypotenuse of a triangle
is no greater a quantity than that of the sum of the lengths of the legs of said triangle. The
third and fourth properties are genuinely more mysterious than the first two, but they are
surprisingly strong statements implying many other important properties of Euclidean space –
the consequences of these will be discussed later. A simplified way of thinking about condition
3, however, is as the statement that the area of a parallelogram is equivalent to that of a rectangle
of the same height and base width. Now, our reason for imposing square-summability on the
coordinates of the analogous vector x “ px1 , x2 , ¨ ¨ ¨ q in `2 pNq forces the quantity
}x} “ lim }x}n “
a
nÑ8
|x1 |2 ` |x2 |2 ` ¨ ¨ ¨
to be finite. As it turns out, the quantity }x} satisfies the four properties of Euclidean distance
above, so we will refer to the elements of `2 pNq as vectors or points.
Consider again the third property of Euclidean distance listed in p1q. For reasons beyond
the scope of this paper, the fact that the Parallelogram Law holds in `2 pNq implies that we may
discuss the orthogonality of vectors in `2 pNq. In other words, given two points X and Y in `2 pNq,
one could in principle measure the ‘angle’ =XOY , where O denotes the origin p0, 0, 0, ¨ ¨ ¨ q.
Measuring angles in the intuitive sense turns out to be mostly unhelpful in the study of `2 pNq,
however, and is rightly replaced by another quantity. For vectors with n coordinates, we define
the quantity
n
ÿ
xx, yyn “
xk yk ,
k“1
where y k denotes the complex conjugate of yk . A helpful example when thinking about the
geometric significance of this quantity occurs when }x}n “ 1: |xx, yyn | is the length of the
vector y ’s projection onto the vector x.
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Three important properties of x¨, ¨y follow directly from the definition given:
1. xx, xyn “ }x}2n ,
(2)
2. xx, yyn “ xy, xyn ,
3. If a, b P C, xax ` by, zy “ axx, zy ` bxy, zy.
The first property is a direct connection between the definition of x¨, ¨yn and } ¨ }n , and in fact,
we could have defined the latter in terms of the former. In our eventual generalization of these
concepts we will do this, as these imply the first three properties listed in (1) via arguments an
intuitionist would accept. For instance, the Parallelogram law follows from
}x ` y}2n ` }x ´ y}2n “ xx ` y, x ` yyn ` xx ´ y, x ´ yyn
“ xx, xyn ` xx, yyn ` xy, xyn ` xy, yyn ` xx, xyn ` xx, ´yyn ` x´y, xyn ` xy, yyn
“ 2xx, xyn ` 2xy, yyn ` xx, yyn ` xy, xyn ´ xx, yyn ´ xy, xyn
“ 2}x}2n ` 2}y}2n .
Now, since the size of xx, yyn is bounded above by the sum of the quantities |xk ||yk |, and
we have required that x and y have square-summable entries, a theorem from [2] called Hölder’s
Inequality implies that there is a natural number M such that for each n the inequality xx, yyn ď
M holds. Again, this implies that we may also extend x¨, ¨yn to a quantity defined to measure
orthogonality in `2 pNq, namely
xx, yy “ lim xx, yyn .
nÑ8
Since this is a classical exposition, confirmation that this limit exists and satisfies the properties
listed in (2) can be found in [2]. Note that I will refer to approximately Euclidean space as a set
equipped with a two place function x¨, ¨y satisfying the properties listed in (2), and by extension
a function } ¨ } satisfying the properties 1 through 3 in (1).
Classically speaking, the above discussion tells us that one way to view `2 pNq is as an infinitedimensional version of Euclidean geometry. Indeed, its most careful investigator, David Hilbert,
certainly saw it this way. But an intuitionist would probably have a few issues with Hilbert’s
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work in this area, since it not only relies on the classical development of real and complex
numbers, but it also rests on the existence of infinite sequences. Below we will develop a similar
enough object which is intuitionistically acceptable, roughly by making the jump from infinite to
potentially infinite sequences while using the intuitionist analog of real and complex numbers.
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Hilbert Space
Classical mathematicians usually call approximately Euclidean spaces Hilbert spaces, and our
constructive definition of Hilbert space is going reflect this. Practically speaking, most classical
Hilbert spaces are structurally similar enough (the differences mostly being set theoretical) to
an intuitionist Hilbert space. In fact, finding a definition of Hilbert space which satisfies the
intuitionist has philosophical significance, especially for physicists: Much of quantum mechanics
depends directly on the the theory of Hilbert spaces, and far as actual physics is concerned,
quantum mechanics asks what are currently the most fundamental scientific questions regarding
physical reality. It would be nice, in the opinion of the quantum physicist, to know that the theory
on which they base their claims does not depend on controversial mathematics, considering how
controversial their field seems to be. This is an important part of what motivates the construction
below.
We will begin with an abstract framework for complex vector spaces, and the construction
of these depend on the concepts of real number generator and species. Suppose that the rational
numbers have been constructed.
Definition 1. A real number generator is a Cauchy sequence of rational numbers generated
by an algorithm. Where tan u and tbn u are real number generators, if an “ bn for every natural
number n, we write tan u ” tbn u and call them identical. If two real number generators
a ” tan u and b ” tbn u satisfy the sentence
ˆ
˙
1
`
@k P Z DN n ą N Ñ |an ´ bn | ă
,
k
we write a “ b and say that a and b coincide. If there exists a k P Z` and an N P N such that
for each n ą N we have
|an ´ bn | ą
1
,
k
we write a#b and say that they lie apart.
Real number generators are the intuitionist’s analog of of real numbers, as they provide
a constructive completion of the rational numbers (completion in a sense similar to that of
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property 4 from (1)). For a more detailed treatment of these concepts, see [5].
As the terminology suggests, equality in the sense described in definition 1 forms an equivalence relation. Notice also that writing a ‰ b for two real number generators a ” tan u and
” tbn u does not necessarily say that a#b, though the converse of this is true. For instance,
let b ” 1 and an be the reciprocal of either the smallest natural number greater than n which
satisfies Goldbach’s conjecture, or 1 if there are none. Certainly a ‰ b, but the mathematical
community has yet to decide whether or not a#b! The terms ‘real number’ and ‘real number
generator’ will be used interchangeably, and one can add and multiply real number generators
just like in the classic field of real numbers. In other words, when a ” tan u and b ” tbn u,
a ` b “ tan u ` tbn u :“ tan ` bn u and a ¨ b “ tan u ¨ tbn u :“ tan ¨ bn u.
are well-defined quantities, and ` and ¨ have the expected properties (a proof for this can be
found in [5]). I will often omit the symbol ¨ from ‘a ¨ b’ for brevity’s sake. The following notion is
one of the intuitionist’s analogs for ‘set’.
Definition 2. A species is a property which a mathematical entity may posess. An element or
member of a species is a mathematical entity for which the species holds. The species of real
number generators is denoted R, and the species of pairs of real number generators is denoted
R2 .
The species R2 when equipped with the product ¨ defined by
pr, sq ¨ pt, uq “ prt ´ su, st ` ruq,
the operation pr, sq “ pr, ´sq, and coordinate-wise addition `, is denoted C. The species
C is the intuitionist’s analog of the complex numbers, and members of this species are called
complex number generators. Two complex number generators pr, sq and pt, uq coincide if and
only if r “ t and s “ u. It will be taken for granted that the complex number generators which
are non-zero have inverses, though this point isn’t important here. Now that we have a species
analogous to the complex numbers at our disposal, we can begin to develop an intuitionistically
acceptable framework on which Hilbert spaces can be built.
Definition 3. A species of vectors denoted V , equipped with the two-place function x¨, ¨y from
V 2 to C, and the two-place function p¨ ` ¨q from V 2 to V , and the two-place function p¨qp¨q
from C ˆ V to V , is called an inner product space when
1. αr ` βs is of the species V whenever α, β are complex number generators and r, s P V ,
and
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2. x¨, ¨y is an inner product. In other words,
a) @r P V , xr, ry is a real number greater than or equal to 0,
b) @r, s P V , @α, β P C, xαr ` βs, ty “ αxr, ty ` βxs, ty, and
c) xr, sy “ xs, ry.
3. If xr ´ s, r ´ sy “ 0, we write r “ s.
4. If xr ´ s, r ´ sy#0, we write r#s.
We define the quantity }r} for each r P V to be
a
xr, r, y, and interpret this as the vector r ’s
length. Note that the symbol 0 will be used to denote the vector 0r , where r is any vector and 0 is
the complex number generator p0, 0q. The repeated use of symbols is intentional: The classical
Euclidean spaces are analogously inner product spaces, as well as `2 pNq. Furthermore, since
every inner product space satisfies properties 1 through 3 in (2), inner product spaces also satisfy
1 through 3 of (1). Not all inner product spaces necessarily look like one of these, however. We
have not demanded that inner product spaces satisfy property 4 of p1q. For instance, a classical
mathematician could discuss the inner product space `c pNq of sequences of complex numbers
whose entries are equal to zero at all but finitely many indices, with the same inner product as
`2 pNq. `c pNq contains the vector
ˆ
sn “
1 1 1
1
1, , , , ¨ ¨ ¨ , n , 0, 0, 0, ¨ ¨ ¨
2 4 8
2
˙
for each natural number n. But, letting j and l be natural numbers, j ą l , and k P Z` , the choice
of N “ k and the geometric series formula give the inequality
` ˘j`1
` ˘l`1
ˆ ˙n
j
ÿ
1 ´ 12
1 ´ 12
1
}sj ´ sl } “
´
“
2
1 ´ 12
1 ´ 12
n“l`1
ˆ ˙l ˜
ˆ ˙j´l ¸ ˆ ˙l ˆ ˙k
1
1
1
1
1
“
1´
ă
ă
ă
2
2
2
2
k
2
whenever l ą k . This shows us that tsn u is a Cauchy sequence in `c pNq. The limit of sn is
the sequence p1{2n q, however, which does not appear in `c pNq since it is non-zero at every
index! The same issue appears in the construction of the intuitionist analog of real numbers, as
Cauchy sequences of rationals do not necessarily converge to rational numbers. We follow in
the footsteps of [5] to fix this issue.
Definition 4. Suppose we have constructed the inner product space V with x¨, ¨y. The species
of sequences in V which are Cauchy with respect to } ¨ } is the Hilbert space V about V . Where
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trn u and tsn u are members of V , if rn “ sn for every natural number n, we write trn u ” tsn u
and call them identical. If two members of V , r ” trn u and s ” tsn u, satisfy the sentence
ˆ
˙
1
@k P Z` DN P N n ą N Ñ }rn ´ sn } ă
,
k
we write r “ s and say that r and s coincide. If there exists a k P Z` and an N P N such that
for each n ą N we have
}rn ´ sn } ą
1
,
k
we write r#s and say that they lie apart.
Setting xtrn u, tsn uy “ txrn , sn yu, trn u ` tsn u “ trn ` sn u, and αtrn u “ tαrn u, turns V
into an inner product space. The proof that these are well-defined operations is similar to the
same proof given for the addition and multiplication of real number generators found in [5].
For those who have seen a classical definition of Hilbert space: This definition is likely
strange, as it seems to imply that a Hilbert space is defined with respect to one of its dense
subspaces. Practically speaking, however, this can be a natural way of thinking about Hilbert
spaces. A technical example of this is the set of square-integrable functions, denoted L2 pRq:
L2 pRq is the completion of the inner product space of continuous functions f : R Ñ C satisfying
ż8
|f pxq|2 dx ă 8,
´8
with the inner product
ż8
xf, gy “
f pxqgpxqdx,
´8
so that we would say that L2 pRq is the Hilbert space about this set. Of course, this example comes
from classical mathematics. The next example is of fundamental importance to intuitionistic
Hilbert space theory, and will conclude our technical discussion.
Suppose we have constructed the object `c pNq, the species of sequences of complex number
generators z ” tzn u such that there is a natural number Nz , dependent on z , such that zn “ 0
for every n ą Nz . In other words, for only finitely many indices n is zn #0. With respect to
coordinate-wise scaling and addition, `c pNq satisfies point 1 of definition 3. Setting z ” tzn u
and w ” twn u, and N “ mintNz , Nw u, we define
xz, wy “
N
ÿ
zn w n .
n“1
It follows readily that `c pNq equipped with x¨, ¨y is an inner product space. Now, to see that the
Hilbert space `c pNq is then the intuitionist analog of the approximately Euclidean space `2 pNq,
it suffices to check that it satisfies property 4 from (1) - this is included as an appendix.
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3
Discussion
A brief reflection on what has been done is in order. The classical notion of Hilbert space (what
we’ve called approximately Euclidean space), in particular `2 pNq, is widely used in physics and
engineering; state spaces of paricles in quantum mechanics and the use of Fourier transforms
in signal processing are some examples. But the notion of Hilbert space is general enough to
encompass many inherently infinitary objects, including `2 pNq (whose elements are themselves
completed infinities). As Hilbert himself learned, the intuitionistic skepticism surrounding
completed infinities is philosophically reasonable – a discomforting fact for some engineers
and physicists, I’m sure. What has been done above is in the spirit of easing this discomfort,
providing a construction of the Hilbert space `2 pNq without the use of completed infinities.
Appendix: `c pNq is the new `2 pNq
Here we sketch a verification that `2 pNq satisfies property 4 of (1). Let
pr pkq qk “ ptrnp1q u, trnp2q u, trnp3q u, ¨ ¨ ¨ q
p1q
p2q
p3q
be a Cauchy sequence in `c pNq, and r “ tr1 , r2 , r3 , ¨ ¨ ¨ u. We first show that r is a member
of the species `c pNq: For any p P Z` we can find an N P N such that
pjq
piq
t}rk ´ rk }uk “ }r piq ´ r pjq } ă
1
2p
piq
for each i, j ą N , since pr pkq qk is Cauchy. Notice that we could have chosen N so that }rk ´
pjq
rk } ă 1{2k for each k ą N as well. Now, since each r pkq P `2 pNq, we can find an M P N such
that i, j ą M implies
pkq
}ri
1
.
2p
pkq
´ rj } ă
Choosing i, j, k ą N ` M , we see that
piq
pjq
piq
piq
piq
pjq
piq
piq
piq
pjq
}ri ´ rj } “ }ri ´ rk ` rk ´ rk } ­ą }ri ´ rk } ` }rk ´ rk } ă
1
.
p
This tells us that r is a Cauchy sequence of elements of `c pNq. Now we check that r is the limit
of the sequence pr pkq qk : Since for any i P N we have
pjq
piq
}r ´ r piq } “ t}rj ´ rj }uj
pjq
piq
pjq
piq
piq
piq
“ t}rj ´ ri ` ri ´ rj }uj
piq
piq
­ą t}rj ´ ri }uj ` t}ri ´ rj }uj ,
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r P `2 pNq, and r piq is Cauchy, for each p P Z` there exists an N P N such that
pjq
piq
t}rj ´ ri }uj ă
1
1
piq
piq
and t}ri ´ rj }uj ă
,
2p
2p
we have
pjq
piq
piq
piq
t}rj ´ ri }uj ` t}ri ´ rj }uj ă
1
.
p
This concludes our verification. It is important to note that r was constructed explicitly, making
the above verification intuitionistically acceptable.
Bibliography
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Bibliography
[1] Intuitionism: An Introduction, Heyting, A., North-Holland (1966).
[2] Real and Complex analysis, Rudin, W., McGraw-Hill (1966).
[3] Intuitionism in the Philosophy of Mathematics, R. Iemhoff, The Stanford Encyclopedia of
Philosophy (Winter 2016 Edition).
Link: https://plato.stanford.edu/cgi-bin/encyclopedia/archinfo.cgi?entry=intuitionism
[4] Natural Topology, Frank Waaldijk, Brouwer Society, Nijmegen, the Netherlands (2012).
Link: http://www.fwaaldijk.nl/natural-topology.pdf
[5] Philosophies of Mathematics, A. L. George, D. Velleman, Wiley (2001)
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