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What would be an example of a not (topologically) closed subspace of a normed space?
mathboy said:R is a normed space, so take any open interval.
lady99 said:why the space of diffrental function not closed help me pleas quakly
A "not closed linear subspace" refers to a subset of a vector space that does not include all of the vectors necessary for it to be considered a closed subspace. In other words, there are some vectors that cannot be reached by taking linear combinations of the vectors within the subspace.
The significance of a "not closed linear subspace" is that it cannot be used to fully describe the vector space it is a subset of. This can cause issues when trying to use linear algebra to solve problems or make predictions based on vectors within the subspace.
A "not closed linear subspace" differs from a closed linear subspace in that it does not contain all of the vectors necessary to form a complete subspace. A closed linear subspace, on the other hand, includes all necessary vectors and can be used to fully describe the vector space it is a part of.
Examples of "not closed linear subspaces" include a line that does not pass through the origin in a two-dimensional space, or a plane that does not pass through the origin in a three-dimensional space. In both cases, there are vectors that cannot be reached by taking linear combinations of the vectors within the subspace.
A "not closed linear subspace" can be identified by checking if it contains all of the necessary vectors to form a complete subspace. This can be done by checking if the subspace contains the zero vector, is closed under vector addition, and is closed under scalar multiplication. If any of these conditions are not met, then the subspace is not closed and is therefore a "not closed linear subspace".