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Here is a supplement to the nice answer that you got at MSE.

Much of the theory of infinite dimensional vector spaces is motivated by solving concrete problems in analysis. To solve differential equations, it is often profitable to use vector spaces of functions, and it is for this purpose that the theory of Banach spaces and other areas of functional analysis were developed. It is no surprise that in an analytic context one is concerned with questions of distance/absolute value, defining infinite sums, different notions of convergence etc.

On the other hand, as noted in Mikhail's response, the algebraic theory of infinite dimensional vector spaces is not particularly interesting on its own. For the most part, one of the following two things happens

1. There is an entirely analogous theory to the finite dimensional case (e.g. there is one isomorphism class of vector space for each cardinality of set; a linear transformation is determined uniquely by its values on a basis; a linear transformation is invertible iff the kernel and cokernel vanish etc.).
2. There is no possible analogous theory to the finite dimensional case, without introducing some notion of convergence/topology (e.g. there aren't infinite dimensional determinants that you can use to detect invertibility).

This makes the algebraic theory less interesting than the analytic one.

In summary, the theory of infinite dimensional vector spaces has an analytic flavor because the historical motivations and applications are analytic, and most of the new nontrivial theory lies in an an analytic direction.

Edit: One final comment: so-called "pure" infinite dimensional vector spaces actually do appear in mathematical practice quite frequently (at least in algebra). But there typically aren't classes or books devoted specifically to them, for the reasons mentioned above.

Here is a supplement to the nice answer that you got at MSE.

Much of the theory of infinite dimensional vector spaces is motivated by solving concrete problems in analysis. To solve differential equations, it is often profitable to use vector spaces of functions, and it is for this purpose that the theory of Banach spaces and other areas of functional analysis were developed. It is no surprise that in an analytic context one is concerned with questions of distance/absolute value, defining infinite sums, different notions of convergence etc.

On the other hand, as noted in Mikhail's response, the algebraic theory of infinite dimensional vector spaces is not particularly interesting on its own. For the most part, one of the following two things happens

1. There is an entirely analogous theory to the finite dimensional case (e.g. there is one isomorphism class of vector space for each cardinality of set; a linear transformation is determined uniquely by its values on a basis; a linear transformation is invertible iff the kernel and cokernel vanish etc.).
2. There is no possible analogous theory to the finite dimensional case, without introducing some notion of convergence/topology (e.g. there aren't infinite dimensional determinants that you can use to detect invertibility).

This makes the algebraic theory less interesting than the analytic one.

In summary, the theory of infinite dimensional vector spaces has an analytic flavor because the historical motivations and applications are analytic, and most of the new nontrivial theory lies in an analytic direction.

Edit: One final comment: so-called "pure" infinite dimensional vector spaces actually do appear in mathematical practice quite frequently (at least in algebra). But there typically aren't classes or books devoted specifically to them, for the reasons mentioned above.

Here is a supplement to the nice answer that you got at MSE.

Much of the theory of infinite dimensional vector spaces is motivated by solving concrete problems in analysis. To solve differential equations, it is often profitable to use vector spaces of functions, and it is for this purpose that the theory of Banach spaces and other areas of functional analysis were developed. It is no surprise that in an analytic context one is concerned with questions of distance/absolute value, defining infinite sums, different notions of convergence etc.

On the other hand, as noted in Mikhail's response, the algebraic theory of infinite dimensional vector spaces is not particularly interesting on its own. For the most part, one of the following two things happens

1. There is an entirely analogous theory to the finite dimensional case (e.g. there is one isomorphism class of vector space for each cardinality of set; a linear transformation is determined uniquely by its values on a basis; a linear transformation is invertible iff the kernel and cokernel vanish etc.).
2. There is no possible analogous theory to the finite dimensional case, without introducing some notion of convergence/topology (e.g. there aren't infinite dimensional determinants that you can use to detect invertibility).

This makes the algebraic theory less interesting than the analytic one.

In summary, the theory of infinite dimensional vector spaces has an analytic flavor because the historical motivations and applications are analytic, and most of the new nontrivial theory lies in an an analytic direction.

Here is a supplement to the nice answer that you got at MSE.

Much of the theory of infinite dimensional vector spaces is motivated by solving concrete problems in analysis. To solve differential equations, it is often profitable to use vector spaces of functions, and it is for this purpose that the theory of Banach spaces and other areas of functional analysis were developed. It is no surprise that in an analytic context one is concerned with questions of distance/absolute value, defining infinite sums, different notions of convergence etc.

On the other hand, as noted in Mikhail's response, the algebraic theory of infinite dimensional vector spaces is not particularly interesting on its own. For the most part, one of the following two things happens

1. There is an entirely analogous theory to the finite dimensional case (e.g. there is one isomorphism class of vector space for each cardinality of set; a linear transformation is determined uniquely by its values on a basis; a linear transformation is invertible iff the kernel and cokernel vanish etc.).
2. There is no possible analogous theory to the finite dimensional case, without introducing some notion of convergence/topology (e.g. there aren't infinite dimensional determinants that you can use to detect invertibility).

This makes the algebraic theory less interesting than the analytic one.

In summary, the theory of infinite dimensional vector spaces has an analytic flavor because the historical motivations and applications are analytic, and most of the new nontrivial theory lies in an an analytic direction.

Edit: One final comment: so-called "pure" infinite dimensional vector spaces actually do appear in mathematical practice quite frequently (at least in algebra). But there typically aren't classes or books devoted specifically to them, for the reasons mentioned above.

Here is a supplement to the nice answer that you got at MSE.

Much of the theory of infinite dimensional vector spaces is motivated by solving concrete problems in analysis. To solve differential equations, it is often profitable to use vector spaces of functions, and it is for this purpose that the theory of Banach spaces and other areas of functional analysis were developed. It is no surprise that in an analytic context one is concerned with questions of distance/absolute value, defining infinite sums, different notions of convergence etc.

On the other hand, as noted in Mikhail's response, the algebraic theory of infinite dimensional vector spaces is not particularly interesting on its own. For the most part, one of the following two things happens

1. There is an entirely analogous theory to the finite dimensional case (e.g. there is one isomorphism class of vector space for each cardinality of set; linear transformations are determined uniquely by the image of a basis; a linear transformation is invertible iff the kernel and cokernel vanish etc.).
2. There is no possible analogous theory to the finite dimensional case, without introducing some notion of convergence/topology (e.g. there aren't infinite dimensional determinants that you can use to detect invertibility).

This makes the algebraic theory less interesting than the analytic one.

In summary, the theory of infinite dimensional vector spaces has an analytic flavor because the historical motivations and applications are analytic, and most of the new nontrivial theory lies in an an analytic direction.

Here is a supplement to the nice answer that you got at MSE.

Much of the theory of infinite dimensional vector spaces is motivated by solving concrete problems in analysis. To solve differential equations, it is often profitable to use vector spaces of functions, and it is for this purpose that the theory of Banach spaces and other areas of functional analysis were developed. It is no surprise that in an analytic context one is concerned with questions of distance/absolute value, defining infinite sums, different notions of convergence etc.

On the other hand, as noted in Mikhail's response, the algebraic theory of infinite dimensional vector spaces is not particularly interesting on its own. For the most part, one of the following two things happens

1. There is an entirely analogous theory to the finite dimensional case (e.g. there is one isomorphism class of vector space for each cardinality of set; a linear transformation is determined uniquely by its values on a basis; a linear transformation is invertible iff the kernel and cokernel vanish etc.).
2. There is no possible analogous theory to the finite dimensional case, without introducing some notion of convergence/topology (e.g. there aren't infinite dimensional determinants that you can use to detect invertibility).

This makes the algebraic theory less interesting than the analytic one.

In summary, the theory of infinite dimensional vector spaces has an analytic flavor because the historical motivations and applications are analytic, and most of the new nontrivial theory lies in an an analytic direction.