Computations with Function Spaces

Reflexive Spaces

Here we reproduce an example from our blog involving symbolic computation on entities. One of the beautiful things about computational encoding (and part of the reason it is so desirable for mathematics as a whole) is that known results can be easily tested or verified (and new ones proposed or tested). As an example, consider the duality of Lebesgue spaces
p
L
and
q
L
for
1/p+1/q1
with
p≥1
. First, define a variable to represent the
p
L
entity:
In[1]:=
lp=
Lebesgue space
p.
L
(
n.

, 
n.
x
)
FUNCTION SPACE
;
Now, use the "DualSpace" property to obtain the dual entity:
In[2]:=
lq=lp
dual space

Out[2]=
Lebesgue space
p./(-1+p.)
L
(
n.

, 
n.
x
)
As can be seen, this formulation allows computation to be performed and expressed through the elegant paradigm of symbolic transformation of the entity canonical name. Taking the dual space of
q
L
in turn then gives:
In[3]:=
lq
dual space

Out[3]=
Lebesgue space
p./(-1+p.)-1+
p.
-1+p.
L
(
n.

, 
n.
x
)
Finally, applying symbolic simplification to the entity canonical name:
In[4]:=
%//Simplify
Out[4]=
Lebesgue space
p.
L
(
n.

, 
n.
x
)
This verifies we have obtained the same space we originally started with:
In[5]:=
%lp
Out[5]=
True
... in other words, that the double dual
**
(
p
L
)
, where * denotes the dual space, is equivalent to
p
L
.
Function spaces with this property are said to be reflexive, so let's now verify that we have correctly categorized Lebesgue spaces among such spaces. Here, we use a Lebesgue
p
L
space with a concrete value of
p≥1
(which is a requirement for reflexivity) for checking the "Classifications" property:
In[6]:=
Lebesgue space
2
L
(
n.

, 
n.
x
)
FUNCTION SPACE
["Classifications"]
Out[6]=

Baire space
,
Banach space
,
barrelled space
,
bornological space
,
compactly generated space
,
complete space
,
convenient space
,
Fréchet space
,
F-space
,
Hilbert space
,
inner product space
,
locally complete space
,
locally convex space
,
Lusin space
,
Mackey space
,
metrizable space
,
normed space
,
Polish space
,
pseudo-complete space
,
pseudo-metrizable space
,
quasi-Banach space
,
quasi-barrelled space
,
quasi-complete space
,
quasi-normed space
,
Radon space
,
reflexive space
,
seminormed space
,
semi-reflexive space
,
separable space
,
sequentially complete space
,
stereotype space
,
Suslin space
,
topological vector space
,
webbed space

Taking the canonical names of these:
In[7]:=
CanonicalName[%]
Out[7]=
{BaireSpace,BanachSpace,BarrelledSpace,BornologicalSpace,CompactlyGeneratedSpace,CompleteSpace,ConvenientSpace,FrechetSpace,FSpace,HilbertSpace,InnerProductSpace,LocallyCompleteSpace,LocallyConvexSpace,LusinSpace,MackeySpace,MetrizableSpace,NormedSpace,PolishSpace,PseudoCompleteSpace,PseudoMetrizableSpace,QuasiBanachSpace,QuasiBarrelledSpace,QuasiCompleteSpace,QuasiNormedSpace,RadonSpace,ReflexiveSpace,SemiNormedSpace,SemiReflexiveSpace,SeparableSpace,SequentiallyCompleteSpace,StereotypeSpace,SuslinSpace,TopologicalVectorSpace,WebbedSpace}
... we now indeed see that this space is reflexive:
In[8]:=
MemberQ[%,"ReflexiveSpace"]
Out[8]=
True
We can easily get a list of all reflexive function spaces in our curated set of function spaces either directly by considering reflexive spaces as an entity in the "TopologicalSpaceType" domain:
In[9]:=
Entity["TopologicalSpaceType","ReflexiveSpace"]["ExampleFunctionSpaces"]
Out[9]=

Bargmann-Segal-Fock space (
n.

,
n.
z
)
,
Bergman space
2
A
(,
2
λ
)
,
Bessel-potential space
s.
H
2
(
n.

,
n.
x
)
,
classical Besov space
s.
Λ
p.,q.
(
n.

, 
n.
x
)
,
continuous space
ω
C
([a., b.])
,
continuous space
ω
C
(
n.

)
,
harmonic Bergman space
2
a
(,
2
λ
)
,
harmonic Bergman space
p.
a
(,
2
λ
)
,
harmonic Hardy space
2
h
()
,
holomorphic Hardy space
2
H
()
,
Lebesgue space
2
L
(,
2
λ
)
,
Lebesgue space
2
L
(
n.

, 
n.
x
)
,
sequence space
2
ℓ
(
+

)
,
Sobolev space
k.
W
2
(
n.

, 
n.
x
)
,
weighted Bergman space
2
A
a.
(,
2
λ
)

Or by creating an implicitly defined entity class of function spaces whose "Classifications" include "ReflexiveSpace":
In[10]:=
EntityClass["FunctionSpace","Classifications"ContainsAny[{"ReflexiveSpace"}]]
Out[10]=
function spaces
Expanding out using EntityList:
In[11]:=
EntityList[%]
Out[11]=

Bargmann-Segal-Fock space (
n.

,
n.
z
)
,
Bergman space
2
A
(,
2
λ
)
,
Bessel-potential space
s.
H
2
(
n.

,
n.
x
)
,
classical Besov space
s.
Λ
p.,q.
(
n.

, 
n.
x
)
,
continuous space
ω
C
([a., b.])
,
continuous space
ω
C
(
n.

)
,
harmonic Bergman space
2
a
(,
2
λ
)
,
harmonic Bergman space
p.
a
(,
2
λ
)
,
harmonic Hardy space
2
h
()
,
holomorphic Hardy space
2
H
()
,
Lebesgue space
2
L
(,
2
λ
)
,
Lebesgue space
2
L
(
n.

, 
n.
x
)
,
sequence space
2
ℓ
(
+

)
,
Sobolev space
k.
W
2
(
n.

, 
n.
x
)
,
weighted Bergman space
2
A
a.
(,
2
λ
)


Norms, Inner Products and the Schwarz Inequality

To illustrate how the symbolic representation of function space properties can be used to perform actual computations, consider the Lebesgue space
2
L
:
In[12]:=
l2=
Lebesgue space
2
L
(
n.

, 
n.
x
)
FUNCTION SPACE
;
Its norm, as expressed using the "Norm" property, involves a number of PureMath` extensions to the Wolfram Language:
In[13]:=
l2norm=l2["Norm"]
Out[13]=
Functionf.,Integrate
2
Abs[f.[x.]]
,{x.,PureMath`MeasureSpace[
n.
Reals
,PureMath`LebesgueMeasure[n.]]}
... in particular, the symbols MeasureSpace and LebesgueMeasure. The Lebesgue measure is an extension of the classical notions of length and area to more complicated sets, and the Lebesgue integral (an integral over a measure space with Lebesgue measure) covers a wider class of functions than does the Wolfram Language's standard Riemann integral.
Consider the simple case of wanting to compute the norm of a univariate real function. In that case, we can write a set of transformations that effectively map the full mathematical result to one that can be evaluated for concrete univariate real functions:
In[14]:=
fromentity[fn_]:=fn/.{​​{x.,PureMath`MeasureSpace[
n_
Reals
,PureMath`LebesgueMeasure[n_]]}{x.,-∞,∞},​​Inactive[Integrate]Integrate,​​Conjugate[f_][x_]:>Conjugate[f[x]]​​}
Applying this transformation function to the norm given above and then to an arbitrary function f gives the unevaluated result:
In[15]:=
fromentity[l2norm][f]
Out[15]=
∞
∫
-∞
2
Abs[f[x.]]
x.
For particular functions, say:
In[16]:=
f1[x_]:=Exp[-x^2]​​f2[x_]:=Piecewise[{{1-x^2,Abs[x]<1},{0,True}}]
These can then be evaluated explicitly using the Wolfram Language's built-in Integrate function:
In[18]:=
fromentity[l2norm][f1]
Out[18]=
1/4
π
2
In[19]:=
fromentity[l2norm][f2]
Out[19]=
4
15
We do the same thing with the inner product, applying the general result:
In[20]:=
l2inner=l2["InnerProduct"]
Out[20]=
Function[{f1,f2},Integrate[f1[x.]Conjugate[f2][x.],{x.,PureMath`MeasureSpace[
n.
Reals
,PureMath`LebesgueMeasure[n.]]}]]
... to our chosen functions:
In[21]:=
l2inner[f1,f2]
Out[21]=
Integrate
-
2
x.

Conjugate[f2][x.],{x.,PureMath`MeasureSpace[
n.
Reals
,PureMath`LebesgueMeasure[n.]]}
... and thus obtaining the result:
In[22]:=
fromentity[%]
Out[22]=
2+
π
Erf[1]
2
We are now in a position to verify that famous Schwarz (or Cauchy–Schwarz) inequality, which states in bra–ket notation that for any two integrable complex functions
ψ
1
(x)
and
ψ
2
(x)
:
2
〈
ψ
1
|
*
ψ
2
〉
≤〈
ψ
1
|
*
ψ
1
〉〈
ψ
2
|
*
ψ
2
〉.
Written out in terms of explicit integrals:
2
∫
ψ
1
(x)
*
ψ
2
(x)x
≤∫
2

ψ
1
(x)
x∫
2

ψ
2
(x)
x
Here is the statement of the inequality returned by the "CauchySchwarzInequality" property:
In[23]:=
EntityValue[l2/.n.1,"CauchySchwarzInequality"][f1,f2]
Out[23]=
Abs[PureMath`InnerProduct[{f1,f2},{{LebesgueL,{{Reals,1},{LebesgueMeasure,1}}},2}]]≤Norm[f1,{{LebesgueL,{{Reals,1},{LebesgueMeasure,1}}},2}]Norm[f2,{{LebesgueL,{{Reals,1},{LebesgueMeasure,1}}},2}]
... which we can verify is indeed satisfied in the specific case of our functions f1 and f2: