What is the difference between linear manifold and linear vector subspace? Does linear manifold need to be closed in summation and multiplication? If it needs to be, then it seems it is the same as a linear vector subspace. However, some people mention in infinite dimension case, they are different.
 A: All the vector spaces and vector subspaces are linear manifolds of any dimension, but not reciprocally, that is, linear manifolds have submanifolds which are not linear.
A: The term “linear manifolds” unfortunately has , as far as I know, at least three different meanings depending on different authors.

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*According Wolfarm MathWorld, it is the same as linear subspaces.


*According to planetmath.org, it is the translation of a linear subspace by a vector v, i.e. affine subspaces.  Example: The line y=x+1 is a linear manifold of R^2 but not a linear subspace as it is the linear subspace y=x shifted by the vector v=(0,1).  Elie Cartan adopted this meaning in his book “The Theory of Spinors”.


*Finally, in an infinite dimensional Banach or Hilbert space, linear manifolds can mean closed linear subspaces, while the term “linear subspaces” is reserved for subspaces that are not necessarily closed.  Here, closed means topologically closed under the topology generated by the norm/inner product.  Example: Consider C[0, 1] —— the space of continuous functions on [0, 1].  The subspace of all polynomials is only a linear subspace of C[0, 1] but not a linear manifold under the sup norm because it is not topologically closed.  John B Conway adopts this meaning in his book “A Course in Functional Analysis”.
