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Let X be a normed vector space. If C is a closed subspace x is a point in X not in C, show that the set C+Fx is closed. (F is the underlying field of the vector space).
A subspace in a normed vector space is a subset of the vector space that is closed under vector addition and scalar multiplication. This means that if we take any two vectors from the subspace and add them together, the resulting vector will also be in the subspace. Similarly, if we multiply a vector from the subspace by a scalar value, the resulting vector will also be in the subspace.
To determine if a subset is a subspace of a normed vector space, we need to check if it satisfies the two conditions of closure under vector addition and scalar multiplication. We can do this by taking any two vectors from the subset and adding them together, and also by multiplying a vector from the subset by a scalar value. If the resulting vectors are also in the subset, then it is a subspace of the normed vector space.
The dimension of a subspace in a normed vector space is the number of linearly independent vectors required to span the subspace. This means that any vector in the subspace can be written as a linear combination of the basis vectors, which are the linearly independent vectors that span the subspace.
Yes, a subspace of a normed vector space can be infinite-dimensional. This means that the number of linearly independent vectors required to span the subspace is infinite. In fact, in some cases, the entire normed vector space can be considered as a subspace of itself.
Some examples of subspaces in a normed vector space include the set of all points on a line or a plane passing through the origin, the set of all polynomials of a certain degree or less, and the set of all continuous functions on a closed interval. Additionally, any vector space itself can be considered as a subspace of itself.