Calculus Let W 1 , W 2 , W 3 , W 4 , and W 5 be defined as in Example 5 . Show that W i is a subspace of W j for i ≤ j . Example 5 Subspaces of Functions (Calculus) Let W 5 be the vector space of all functions defined on [ 0 , 1 ] , and let W 1 , W 2 , W 3 , and W 4 be defined as shown below. W 1 = set of all polynomial functions that are defined on [ 0 , 1 ] W 2 = set of all functions that are differentiable on [ 0 , 1 ] W 3 = set of all functions that are continuous on [ 0 , 1 ] W 4 = set of all functions that are integrable on [ 0 , 1 ] Show that W 1 ⊂ W 2 ⊂ W 3 ⊂ W 4 ⊂ W 5 and that W i is a subspace of W j for i ≤ j .
Calculus Let W 1 , W 2 , W 3 , W 4 , and W 5 be defined as in Example 5 . Show that W i is a subspace of W j for i ≤ j . Example 5 Subspaces of Functions (Calculus) Let W 5 be the vector space of all functions defined on [ 0 , 1 ] , and let W 1 , W 2 , W 3 , and W 4 be defined as shown below. W 1 = set of all polynomial functions that are defined on [ 0 , 1 ] W 2 = set of all functions that are differentiable on [ 0 , 1 ] W 3 = set of all functions that are continuous on [ 0 , 1 ] W 4 = set of all functions that are integrable on [ 0 , 1 ] Show that W 1 ⊂ W 2 ⊂ W 3 ⊂ W 4 ⊂ W 5 and that W i is a subspace of W j for i ≤ j .
Solution Summary: The author illustrates how the set W_i is a subspace of wj if the two closure conditions holds true.
Calculus Let
W
1
,
W
2
,
W
3
,
W
4
, and
W
5
be defined as in Example
5
. Show that
W
i
is a subspace of
W
j
for
i
≤
j
.
Example 5 Subspaces of Functions (Calculus)
Let
W
5
be the vector space of all functions defined on
[
0
,
1
]
, and let
W
1
,
W
2
,
W
3
,
and
W
4
be defined as shown below.
W
1
=
set of all polynomial functions that are defined on
[
0
,
1
]
W
2
=
set of all functions that are differentiable on
[
0
,
1
]
W
3
=
set of all functions that are continuous on
[
0
,
1
]
W
4
=
set of all functions that are integrable on
[
0
,
1
]
Show that
W
1
⊂
W
2
⊂
W
3
⊂
W
4
⊂
W
5
and that
W
i
is a subspace of
W
j
for
i
≤
j
.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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