Are you asking a question here or are you simply informing us? We can't help you if we don't know what you are asking about!nwl said:Let X=C[O,1] and Y=span(x\_{0},x\_{1},···), where X\_{j}={t}^{i}, so that Y is the set of all polynomials. Y is not closed in X.
An infinite dimensional subspace is a subset of a vector space that contains an infinite number of vectors. This means that the subspace is not limited to a finite number of dimensions, unlike a finite dimensional subspace.
An infinite dimensional subspace has an infinite number of basis vectors, while a finite dimensional subspace has a finite number of basis vectors. This means that an infinite dimensional subspace can have an infinite number of linearly independent vectors, while a finite dimensional subspace can only have a limited number of linearly independent vectors.
An infinite dimensional subspace can have a limit point that is not contained within the subspace. This means that the subspace is not closed, as it does not contain all of its limit points. This is different from a finite dimensional subspace, which is always closed.
A limit point in a vector space is a point that can be approached arbitrarily closely by a sequence of points in the space. In other words, a limit point is a point that is the limit of a convergent sequence of points in the space.
Yes, it is possible for an infinite dimensional subspace to be closed. This occurs when the subspace contains all of its limit points. However, this is not always the case, as an infinite dimensional subspace can also have limit points that are not contained within the subspace, making it not closed.