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Real Analysis
Contents
Uniform Continuity
Riemann Integral
Lebesgue Integration
Stone–Weierstrass Theorem
Norm
Sequence
Recurrence Relation
References
Uniform Continuity
Riemann Integral
[u, v] ≤ E, g : [u, v] → R,
(
a ≤ g
(
t
)
≤ b
)
→ a
(
v − u
)
≤
I
[u, v]
g ≤ b
(
v − u
)
.
[a, b] ≤ E, g : [u, v] → R,
(
a ≤ c ≤ b
)
→
I
[a, c]
(
g
)
+
I
[c, b]
(
g
)
=
I
[a, b]
(
g
)
.
g
(
x
)
(
x − x
)
≤
I
[x, x]
(
g
)
≤ g
(
x
)
(
x − x
)
or
I
[x, x]
(
g
)
= 0.
I
[a, b]
(
g
)
= −
I
[b, a]
(
g
)
.
Lebesgue Integration
Stone–Weierstrass Theorem
Norm
a, b : L
(
F
)
, c : F,
p
(
a + b
)
≤ p
(
a
)
+ p
(
b
)
(subadditive, triangle inequality).
p
(
c a
)
= |c| p
(
a
)
(absolutely homogeneous).
(
p
(
a
)
= 0
)
→
(
a = 0
)
(positive definite or point-separating).
Sequence
Recurrence Relation
References
Mark Bridger. Real Analysis: A Constructive Approach Through Interval Arithmetic. 2019.
Institute for Advanced Study. Homotopy Type Theory: Univalent Foundations of Mathematics. 2013.
