Mixing
How can sequences satisfy vector space properties? If sequence is not a vector.
$begingroup$
How can sequences satisfy vector space properties? If sequence is not a vector.
An example semantic problem occurs, when one needs to find a zero element or zero $bar{0}$. In order to do this for sequences one'd intuitively define ${a_n}, a_n=0 forall n$ as the zero element. However, this is a sequence, not a vector.
As to why the treating of sequences as vectors may not be intuitive:
What is the difference between operators, functions, sequences and vectors?
functional-analysis
asked Jan 11 at 10:58
mavaviljmavavilj
2,76211036
$endgroup$
|
show 17 more comments
$begingroup$
How can sequences satisfy vector space properties? If sequence is not a vector.
An example semantic problem occurs, when one needs to find a zero element or zero $bar{0}$. In order to do this for sequences one'd intuitively define ${a_n}, a_n=0 forall n$ as the zero element. However, this is a sequence, not a vector.
As to why the treating of sequences as vectors may not be intuitive:
What is the difference between operators, functions, sequences and vectors?
functional-analysis
asked Jan 11 at 10:58
mavaviljmavavilj
2,76211036
$endgroup$
1
$begingroup$
This is too vague. One sequence is just a single object, so it can't be a vector space (or at least not a very interesting one). It could, of course, be an element in a vector space.
$endgroup$
– lulu
Jan 11 at 11:00
3
$begingroup$
Sure they do. ${a_n}+{b_n}={a_n+b_n}$ and $lambda times {a_n}={lambda a_n}$.
$endgroup$
– lulu
Jan 11 at 11:02
1
$begingroup$
i just wrote out the formal definition. Where's the problem?
$endgroup$
– lulu
Jan 11 at 11:03
1
$begingroup$
I'm really not sure what you are after here. If you have a field $mathbb F$, then the set of sequences of elements in $mathbb F$ form a vector space in an obvious way. Any such sequence is then, of course, a vector in that vector space.
$endgroup$
– lulu
Jan 11 at 11:07
1
$begingroup$
An element of a vector space is called a vector. It may or may not have any resemblance to 'vectors'
$endgroup$
– Shubham Johri
Jan 11 at 11:13
|
show 17 more comments
$begingroup$
How can sequences satisfy vector space properties? If sequence is not a vector.
An example semantic problem occurs, when one needs to find a zero element or zero $bar{0}$. In order to do this for sequences one'd intuitively define ${a_n}, a_n=0 forall n$ as the zero element. However, this is a sequence, not a vector.
As to why the treating of sequences as vectors may not be intuitive:
What is the difference between operators, functions, sequences and vectors?
functional-analysis
asked Jan 11 at 10:58
mavaviljmavavilj
2,76211036
$endgroup$
How can sequences satisfy vector space properties? If sequence is not a vector.
An example semantic problem occurs, when one needs to find a zero element or zero $bar{0}$. In order to do this for sequences one'd intuitively define ${a_n}, a_n=0 forall n$ as the zero element. However, this is a sequence, not a vector.
As to why the treating of sequences as vectors may not be intuitive:
What is the difference between operators, functions, sequences and vectors?
functional-analysis
functional-analysis
asked Jan 11 at 10:58
mavaviljmavavilj
2,76211036
asked Jan 11 at 10:58
mavaviljmavavilj
2,76211036
edited Jan 11 at 11:34
mavavilj
asked Jan 11 at 10:58
mavaviljmavavilj
2,76211036
asked Jan 11 at 10:58
mavaviljmavavilj
2,76211036
asked Jan 11 at 10:58
mavaviljmavavilj
2,76211036
2,76211036
1
$begingroup$
This is too vague. One sequence is just a single object, so it can't be a vector space (or at least not a very interesting one). It could, of course, be an element in a vector space.
$endgroup$
– lulu
Jan 11 at 11:00
3
$begingroup$
Sure they do. ${a_n}+{b_n}={a_n+b_n}$ and $lambda times {a_n}={lambda a_n}$.
$endgroup$
– lulu
Jan 11 at 11:02
1
$begingroup$
i just wrote out the formal definition. Where's the problem?
$endgroup$
– lulu
Jan 11 at 11:03
1
$begingroup$
I'm really not sure what you are after here. If you have a field $mathbb F$, then the set of sequences of elements in $mathbb F$ form a vector space in an obvious way. Any such sequence is then, of course, a vector in that vector space.
$endgroup$
– lulu
Jan 11 at 11:07
1
$begingroup$
An element of a vector space is called a vector. It may or may not have any resemblance to 'vectors'
$endgroup$
– Shubham Johri
Jan 11 at 11:13
|
show 17 more comments
1
$begingroup$
This is too vague. One sequence is just a single object, so it can't be a vector space (or at least not a very interesting one). It could, of course, be an element in a vector space.
$endgroup$
– lulu
Jan 11 at 11:00
3
$begingroup$
Sure they do. ${a_n}+{b_n}={a_n+b_n}$ and $lambda times {a_n}={lambda a_n}$.
$endgroup$
– lulu
Jan 11 at 11:02
1
$begingroup$
i just wrote out the formal definition. Where's the problem?
$endgroup$
– lulu
Jan 11 at 11:03
1
$begingroup$
I'm really not sure what you are after here. If you have a field $mathbb F$, then the set of sequences of elements in $mathbb F$ form a vector space in an obvious way. Any such sequence is then, of course, a vector in that vector space.
$endgroup$
– lulu
Jan 11 at 11:07
1
$begingroup$
An element of a vector space is called a vector. It may or may not have any resemblance to 'vectors'
$endgroup$
– Shubham Johri
Jan 11 at 11:13
1
1
$begingroup$
This is too vague. One sequence is just a single object, so it can't be a vector space (or at least not a very interesting one). It could, of course, be an element in a vector space.
$endgroup$
– lulu
Jan 11 at 11:00
$begingroup$
This is too vague. One sequence is just a single object, so it can't be a vector space (or at least not a very interesting one). It could, of course, be an element in a vector space.
$endgroup$
– lulu
Jan 11 at 11:00
3
3
$begingroup$
Sure they do. ${a_n}+{b_n}={a_n+b_n}$ and $lambda times {a_n}={lambda a_n}$.
$endgroup$
– lulu
Jan 11 at 11:02
$begingroup$
Sure they do. ${a_n}+{b_n}={a_n+b_n}$ and $lambda times {a_n}={lambda a_n}$.
$endgroup$
– lulu
Jan 11 at 11:02
1
1
$begingroup$
i just wrote out the formal definition. Where's the problem?
$endgroup$
– lulu
Jan 11 at 11:03
$begingroup$
i just wrote out the formal definition. Where's the problem?
$endgroup$
– lulu
Jan 11 at 11:03
1
1
$begingroup$
I'm really not sure what you are after here. If you have a field $mathbb F$, then the set of sequences of elements in $mathbb F$ form a vector space in an obvious way. Any such sequence is then, of course, a vector in that vector space.
$endgroup$
– lulu
Jan 11 at 11:07
$begingroup$
I'm really not sure what you are after here. If you have a field $mathbb F$, then the set of sequences of elements in $mathbb F$ form a vector space in an obvious way. Any such sequence is then, of course, a vector in that vector space.
$endgroup$
– lulu
Jan 11 at 11:07
1
1
$begingroup$
An element of a vector space is called a vector. It may or may not have any resemblance to 'vectors'
$endgroup$
– Shubham Johri
Jan 11 at 11:13
$begingroup$
An element of a vector space is called a vector. It may or may not have any resemblance to 'vectors'
$endgroup$
– Shubham Johri
Jan 11 at 11:13
|
show 17 more comments
2 Answers
2
active
oldest
votes
$begingroup$
You should carefully read the accepted answer of the question you linked to. That question was based on misconceptions, and it seems yours is too.
Consider the following passage from that answer:
If the terms of a sequence are elements of a field $F,$ then that sequence is an element of the vector space, over $F,$ of all sequences with terms in $F,$ so it is a 'vector' in that sense.
The comments on that answer indicate that the problem is pedagogical: the early levels of instruction in math (which is all most people ever see)
give only limited perspectives into the field of mathematics.
It is likely that someone will see only finite-dimensional vector spaces,
or at least that only the finite-dimensional vector spaces will be identified as vector spaces.
The more general, advanced viewpoint is that anything that satisfies the formal definition of a vector space is a vector space, and its members are vectors.
There are various ways to state a formal definition of vector space
(here is an example),
but they generally admit the same models of vector spaces,
for example sequences of elements of a field $F$ as described in the passage above.
This may not be intuitive at first. Part of an education in mathematics is gaining experience with the application of the definition of the term vector space
until applications like this become intuitive.
answered Jan 11 at 13:02
David KDavid K
53.9k342116
$endgroup$
$begingroup$
The confusion may not come from "elementary mathematics" but instead from "physics", where the term vector is used with their special connotations. Even in more advanced physics, when they use a Hilbert space, its elements are not called "vectors" but something else, such as "states".
$endgroup$
– GEdgar
Jan 11 at 13:09
$begingroup$
@GEdgar True, and reading even further in the discussion of the earlier question, even some fields of higher mathematics differ in how they use certain terms. So it really would depend on which communities one has been exposed to. On the other hand, after one has seen definitions get broadened or narrowed a few times I suspect it gets easier to get past stumbling blocks such as the one we seem to have here (whatever its exact nature is).
$endgroup$
– David K
Jan 11 at 13:27
add a comment |
$begingroup$
A similar confusion arises in Toplogy. Given any topology on a given set such as the real numbers, then the members of that topology are called "open" by definition of toplogical space. However, unlike open intervals in the usual topology of the real numbers, there is nothing "open", except the name, about the open sets of a topological space. It is simply a matter of terminology, not ontology.
In the same way, given any vector space, the elements of the space as called "vectors" by definition of vector space. There need not be any connection with what is usually called a vector in Euclidean spaces. The name "vector" here merely means that the object is a member of a particular set which is equipped with the operations of vector addition and scalar multiplication and the operations satisfy certain properties. It is entirely possible that the objects of the vector space may have different operations and names in another context. This does not contradict their status as vectors also.
answered Jan 11 at 14:01
SomosSomos
13.6k11235
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3069714%2fhow-can-sequences-satisfy-vector-space-properties-if-sequence-is-not-a-vector%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You should carefully read the accepted answer of the question you linked to. That question was based on misconceptions, and it seems yours is too.
Consider the following passage from that answer:
If the terms of a sequence are elements of a field $F,$ then that sequence is an element of the vector space, over $F,$ of all sequences with terms in $F,$ so it is a 'vector' in that sense.
The comments on that answer indicate that the problem is pedagogical: the early levels of instruction in math (which is all most people ever see)
give only limited perspectives into the field of mathematics.
It is likely that someone will see only finite-dimensional vector spaces,
or at least that only the finite-dimensional vector spaces will be identified as vector spaces.
The more general, advanced viewpoint is that anything that satisfies the formal definition of a vector space is a vector space, and its members are vectors.
There are various ways to state a formal definition of vector space
(here is an example),
but they generally admit the same models of vector spaces,
for example sequences of elements of a field $F$ as described in the passage above.
This may not be intuitive at first. Part of an education in mathematics is gaining experience with the application of the definition of the term vector space
until applications like this become intuitive.
answered Jan 11 at 13:02
David KDavid K
53.9k342116
$endgroup$
$begingroup$
The confusion may not come from "elementary mathematics" but instead from "physics", where the term vector is used with their special connotations. Even in more advanced physics, when they use a Hilbert space, its elements are not called "vectors" but something else, such as "states".
$endgroup$
– GEdgar
Jan 11 at 13:09
$begingroup$
@GEdgar True, and reading even further in the discussion of the earlier question, even some fields of higher mathematics differ in how they use certain terms. So it really would depend on which communities one has been exposed to. On the other hand, after one has seen definitions get broadened or narrowed a few times I suspect it gets easier to get past stumbling blocks such as the one we seem to have here (whatever its exact nature is).
$endgroup$
– David K
Jan 11 at 13:27
add a comment |
$begingroup$
You should carefully read the accepted answer of the question you linked to. That question was based on misconceptions, and it seems yours is too.
Consider the following passage from that answer:
If the terms of a sequence are elements of a field $F,$ then that sequence is an element of the vector space, over $F,$ of all sequences with terms in $F,$ so it is a 'vector' in that sense.
The comments on that answer indicate that the problem is pedagogical: the early levels of instruction in math (which is all most people ever see)
give only limited perspectives into the field of mathematics.
It is likely that someone will see only finite-dimensional vector spaces,
or at least that only the finite-dimensional vector spaces will be identified as vector spaces.
The more general, advanced viewpoint is that anything that satisfies the formal definition of a vector space is a vector space, and its members are vectors.
There are various ways to state a formal definition of vector space
(here is an example),
but they generally admit the same models of vector spaces,
for example sequences of elements of a field $F$ as described in the passage above.
This may not be intuitive at first. Part of an education in mathematics is gaining experience with the application of the definition of the term vector space
until applications like this become intuitive.
answered Jan 11 at 13:02
David KDavid K
53.9k342116
$endgroup$
$begingroup$
The confusion may not come from "elementary mathematics" but instead from "physics", where the term vector is used with their special connotations. Even in more advanced physics, when they use a Hilbert space, its elements are not called "vectors" but something else, such as "states".
$endgroup$
– GEdgar
Jan 11 at 13:09
$begingroup$
@GEdgar True, and reading even further in the discussion of the earlier question, even some fields of higher mathematics differ in how they use certain terms. So it really would depend on which communities one has been exposed to. On the other hand, after one has seen definitions get broadened or narrowed a few times I suspect it gets easier to get past stumbling blocks such as the one we seem to have here (whatever its exact nature is).
$endgroup$
– David K
Jan 11 at 13:27
add a comment |
$begingroup$
You should carefully read the accepted answer of the question you linked to. That question was based on misconceptions, and it seems yours is too.
Consider the following passage from that answer:
If the terms of a sequence are elements of a field $F,$ then that sequence is an element of the vector space, over $F,$ of all sequences with terms in $F,$ so it is a 'vector' in that sense.
The comments on that answer indicate that the problem is pedagogical: the early levels of instruction in math (which is all most people ever see)
give only limited perspectives into the field of mathematics.
It is likely that someone will see only finite-dimensional vector spaces,
or at least that only the finite-dimensional vector spaces will be identified as vector spaces.
The more general, advanced viewpoint is that anything that satisfies the formal definition of a vector space is a vector space, and its members are vectors.
There are various ways to state a formal definition of vector space
(here is an example),
but they generally admit the same models of vector spaces,
for example sequences of elements of a field $F$ as described in the passage above.
This may not be intuitive at first. Part of an education in mathematics is gaining experience with the application of the definition of the term vector space
until applications like this become intuitive.
answered Jan 11 at 13:02
David KDavid K
53.9k342116
$endgroup$
You should carefully read the accepted answer of the question you linked to. That question was based on misconceptions, and it seems yours is too.
Consider the following passage from that answer:
If the terms of a sequence are elements of a field $F,$ then that sequence is an element of the vector space, over $F,$ of all sequences with terms in $F,$ so it is a 'vector' in that sense.
The comments on that answer indicate that the problem is pedagogical: the early levels of instruction in math (which is all most people ever see)
give only limited perspectives into the field of mathematics.
It is likely that someone will see only finite-dimensional vector spaces,
or at least that only the finite-dimensional vector spaces will be identified as vector spaces.
The more general, advanced viewpoint is that anything that satisfies the formal definition of a vector space is a vector space, and its members are vectors.
There are various ways to state a formal definition of vector space
(here is an example),
but they generally admit the same models of vector spaces,
for example sequences of elements of a field $F$ as described in the passage above.
This may not be intuitive at first. Part of an education in mathematics is gaining experience with the application of the definition of the term vector space
until applications like this become intuitive.
answered Jan 11 at 13:02
David KDavid K
53.9k342116
answered Jan 11 at 13:02
David KDavid K
53.9k342116
answered Jan 11 at 13:02
David KDavid K
53.9k342116
answered Jan 11 at 13:02
David KDavid K
53.9k342116
53.9k342116
$begingroup$
The confusion may not come from "elementary mathematics" but instead from "physics", where the term vector is used with their special connotations. Even in more advanced physics, when they use a Hilbert space, its elements are not called "vectors" but something else, such as "states".
$endgroup$
– GEdgar
Jan 11 at 13:09
$begingroup$
@GEdgar True, and reading even further in the discussion of the earlier question, even some fields of higher mathematics differ in how they use certain terms. So it really would depend on which communities one has been exposed to. On the other hand, after one has seen definitions get broadened or narrowed a few times I suspect it gets easier to get past stumbling blocks such as the one we seem to have here (whatever its exact nature is).
$endgroup$
– David K
Jan 11 at 13:27
add a comment |
$begingroup$
The confusion may not come from "elementary mathematics" but instead from "physics", where the term vector is used with their special connotations. Even in more advanced physics, when they use a Hilbert space, its elements are not called "vectors" but something else, such as "states".
$endgroup$
– GEdgar
Jan 11 at 13:09
$begingroup$
@GEdgar True, and reading even further in the discussion of the earlier question, even some fields of higher mathematics differ in how they use certain terms. So it really would depend on which communities one has been exposed to. On the other hand, after one has seen definitions get broadened or narrowed a few times I suspect it gets easier to get past stumbling blocks such as the one we seem to have here (whatever its exact nature is).
$endgroup$
– David K
Jan 11 at 13:27
$begingroup$
The confusion may not come from "elementary mathematics" but instead from "physics", where the term vector is used with their special connotations. Even in more advanced physics, when they use a Hilbert space, its elements are not called "vectors" but something else, such as "states".
$endgroup$
– GEdgar
Jan 11 at 13:09
$begingroup$
The confusion may not come from "elementary mathematics" but instead from "physics", where the term vector is used with their special connotations. Even in more advanced physics, when they use a Hilbert space, its elements are not called "vectors" but something else, such as "states".
$endgroup$
– GEdgar
Jan 11 at 13:09
$begingroup$
@GEdgar True, and reading even further in the discussion of the earlier question, even some fields of higher mathematics differ in how they use certain terms. So it really would depend on which communities one has been exposed to. On the other hand, after one has seen definitions get broadened or narrowed a few times I suspect it gets easier to get past stumbling blocks such as the one we seem to have here (whatever its exact nature is).
$endgroup$
– David K
Jan 11 at 13:27
$begingroup$
@GEdgar True, and reading even further in the discussion of the earlier question, even some fields of higher mathematics differ in how they use certain terms. So it really would depend on which communities one has been exposed to. On the other hand, after one has seen definitions get broadened or narrowed a few times I suspect it gets easier to get past stumbling blocks such as the one we seem to have here (whatever its exact nature is).
$endgroup$
– David K
Jan 11 at 13:27
add a comment |
$begingroup$
A similar confusion arises in Toplogy. Given any topology on a given set such as the real numbers, then the members of that topology are called "open" by definition of toplogical space. However, unlike open intervals in the usual topology of the real numbers, there is nothing "open", except the name, about the open sets of a topological space. It is simply a matter of terminology, not ontology.
In the same way, given any vector space, the elements of the space as called "vectors" by definition of vector space. There need not be any connection with what is usually called a vector in Euclidean spaces. The name "vector" here merely means that the object is a member of a particular set which is equipped with the operations of vector addition and scalar multiplication and the operations satisfy certain properties. It is entirely possible that the objects of the vector space may have different operations and names in another context. This does not contradict their status as vectors also.
answered Jan 11 at 14:01
SomosSomos
13.6k11235
$endgroup$
add a comment |
$begingroup$
A similar confusion arises in Toplogy. Given any topology on a given set such as the real numbers, then the members of that topology are called "open" by definition of toplogical space. However, unlike open intervals in the usual topology of the real numbers, there is nothing "open", except the name, about the open sets of a topological space. It is simply a matter of terminology, not ontology.
In the same way, given any vector space, the elements of the space as called "vectors" by definition of vector space. There need not be any connection with what is usually called a vector in Euclidean spaces. The name "vector" here merely means that the object is a member of a particular set which is equipped with the operations of vector addition and scalar multiplication and the operations satisfy certain properties. It is entirely possible that the objects of the vector space may have different operations and names in another context. This does not contradict their status as vectors also.
answered Jan 11 at 14:01
SomosSomos
13.6k11235
$endgroup$
add a comment |
$begingroup$
A similar confusion arises in Toplogy. Given any topology on a given set such as the real numbers, then the members of that topology are called "open" by definition of toplogical space. However, unlike open intervals in the usual topology of the real numbers, there is nothing "open", except the name, about the open sets of a topological space. It is simply a matter of terminology, not ontology.
In the same way, given any vector space, the elements of the space as called "vectors" by definition of vector space. There need not be any connection with what is usually called a vector in Euclidean spaces. The name "vector" here merely means that the object is a member of a particular set which is equipped with the operations of vector addition and scalar multiplication and the operations satisfy certain properties. It is entirely possible that the objects of the vector space may have different operations and names in another context. This does not contradict their status as vectors also.
answered Jan 11 at 14:01
SomosSomos
13.6k11235
$endgroup$
A similar confusion arises in Toplogy. Given any topology on a given set such as the real numbers, then the members of that topology are called "open" by definition of toplogical space. However, unlike open intervals in the usual topology of the real numbers, there is nothing "open", except the name, about the open sets of a topological space. It is simply a matter of terminology, not ontology.
In the same way, given any vector space, the elements of the space as called "vectors" by definition of vector space. There need not be any connection with what is usually called a vector in Euclidean spaces. The name "vector" here merely means that the object is a member of a particular set which is equipped with the operations of vector addition and scalar multiplication and the operations satisfy certain properties. It is entirely possible that the objects of the vector space may have different operations and names in another context. This does not contradict their status as vectors also.
answered Jan 11 at 14:01
SomosSomos
13.6k11235
answered Jan 11 at 14:01
SomosSomos
13.6k11235
answered Jan 11 at 14:01
SomosSomos
13.6k11235
answered Jan 11 at 14:01
SomosSomos
13.6k11235
13.6k11235
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3069714%2fhow-can-sequences-satisfy-vector-space-properties-if-sequence-is-not-a-vector%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
$begingroup$
This is too vague. One sequence is just a single object, so it can't be a vector space (or at least not a very interesting one). It could, of course, be an element in a vector space.
$endgroup$
– lulu
Jan 11 at 11:00
3
$begingroup$
Sure they do. ${a_n}+{b_n}={a_n+b_n}$ and $lambda times {a_n}={lambda a_n}$.
$endgroup$
– lulu
Jan 11 at 11:02
1
$begingroup$
i just wrote out the formal definition. Where's the problem?
$endgroup$
– lulu
Jan 11 at 11:03
1
$begingroup$
I'm really not sure what you are after here. If you have a field $mathbb F$, then the set of sequences of elements in $mathbb F$ form a vector space in an obvious way. Any such sequence is then, of course, a vector in that vector space.
$endgroup$
– lulu
Jan 11 at 11:07
1
$begingroup$
An element of a vector space is called a vector. It may or may not have any resemblance to 'vectors'
$endgroup$
– Shubham Johri
Jan 11 at 11:13