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Miklós Schweitzer
1993 Miklós Schweitzer
7
analysis
analysis
Source: miklos schweitzer 1993 q7
October 22, 2021
Hilbert Space
Functional Analysis
Problem Statement
Let H be a Hilbert space over the field of real numbers
R
\Bbb R
R
. Find all
f
:
H
→
R
f: H \to \Bbb R
f
:
H
→
R
continuous functions for which
f
(
x
+
y
+
π
z
)
+
f
(
x
+
2
z
)
+
f
(
y
+
2
z
)
+
f
(
π
z
)
f(x + y + \pi z) + f(x + \sqrt{2} z) + f(y + \sqrt{2} z) + f (\pi z)
f
(
x
+
y
+
π
z
)
+
f
(
x
+
2
z
)
+
f
(
y
+
2
z
)
+
f
(
π
z
)
=
f
(
x
+
y
+
2
z
)
+
f
(
x
+
π
z
)
+
f
(
y
+
π
z
)
+
f
(
2
z
)
= f(x + y + \sqrt{2} z) + f (x + \pi z) + f (y + \pi z) + f(\sqrt{2} z)
=
f
(
x
+
y
+
2
z
)
+
f
(
x
+
π
z
)
+
f
(
y
+
π
z
)
+
f
(
2
z
)
is satisfied for any
x
,
y
,
z
∈
H
x , y , z \in H
x
,
y
,
z
∈
H
.
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