1.212
Let
𝑆
be an open set according to the
∥⋅∥
𝑎
and let
x
0
be a point in
𝑆
. Since
𝑆
is
open, it contains an open ball in the
∥⋅∥
𝑎
topology about
x
0
, namely
𝐵
𝑎
(
x
0
, 𝑟
) =
{
x
∈
𝑋
:
∥
x
−
x
0
∥
𝑎
< 𝑟
} ⊆
𝑆
Let
𝐵
𝑏
(
x
0
, 𝑟
) =
{
x
∈
𝑋
:
∥
x
−
x
0
∥
𝑏
< 𝑟
}
be the open ball about
x
0
in the
∥⋅∥
𝑏
topology.
The condition (1.15) implies that
𝐵
𝑏
(
x
0
, 𝑟
)
⊆
𝐵
𝑎
(
x
0
, 𝑟
)
⊆
𝑆
and therefore
x
0
∈
𝐵
𝑏
(
x
0
, 𝑟
)
⊂
𝑆
𝑆
is open in the
∥⋅∥
𝑏
topology. Similarly, any
𝑆
open in the
∥⋅∥
𝑏
topology is open in
the
∥⋅∥
𝑎
topology.
1.213
Let
𝑋
be a normed linear space of dimension
𝑛
.
and let
{
x
1
,
x
2
, . . . ,
x
𝑛
}
be
a basis for
𝑋
.
Let
∥⋅∥
𝑎
and
∥⋅∥
𝑏
be two norms on
𝑋
.
Every
x
∈
𝑋
has a unique
representation
x
=
𝛼
1
x
1
+
𝛼
2
x
2
+
⋅ ⋅ ⋅
+
𝛼
𝑛
x
𝑛
Repeated application of the triangle inequality gives
∥
x
∥
𝑎
=
∥
𝛼
1
x
1
+
𝛼
2
x
2
+
⋅ ⋅ ⋅
+
𝛼
𝑛
x
𝑛
∥
𝑎
≤
𝑛
∑
𝑖
=1
∥
𝛼
𝑖
x
𝑖
∥
𝑎
=
𝑛
∑
𝑖
=1
∣
𝛼
𝑖
∣ ∥
x
𝑖
∥
𝑎
≤
𝑘
𝑛
∑
𝑖
=1
∣
𝛼
𝑖
∣
where
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 Fall '10
 Dr.DuMond
 Macroeconomics, Topology, Norm, Metric space, Topological space, Foundations of Mathematical Economics

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