Subspace of Normed Vector Space

In summary: However, the normed space condition is a more fundamental requirement, so it's worth mentioning first.From the definition of convergent sequences, it follows that a_n\rightarrow a+\lambda x\in C+Fx for all n\in N. This means that the limit exists and is in the space.
  • #1
iknowone
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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).
 
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  • #2
Hi there!

Here's my suggestion, but it needs an overview from someone familiar with functional analysis, to be sure it's correct or incorrect:

As X is a normed space, C and Fx are also normed as subspaces of a normed space. So the convergence in these subspaces is set with respect to the original norm of X.

Let [tex]c_n, n\in N;c_n\longwrightarrow\c,c\in C[/tex] be a convergent sequence in C+Fx with limit c. Then, following the structure of the space (because x does not lie in C, C+Fx is a direct sum), we could write [tex]c_n=a_n+\lambda_n x; a_n\in C,\lambda_n\in F[/tex] both convergent sequences [tex]a_n\rightarrow a ;\lambda_n\rightarrow\lambda[/tex] (otherwise c_n could not be convergent).

Since C is closed then [tex]a\in C[/tex] and (assuming F is closed or complete)*** [tex]\lambda\in F[/tex]. So the limit takes the form [tex]c_n\rightarrow a+\lambda x\in C+Fx[/tex] and therefore lies in the space. *** I am not sure, but I think F must also be given as closed. (as a counter example consider the space R^2 over the rationals Q: Set C to be the disc around the origin (closed) and x be some point not in C)

I hope this proof is true at least to some extend.

[edit: spelling :)]
 
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  • #3
Marin is right, you need [itex]F[/itex] to be a complete field, for example the reals or complexes. Otherwise, from [itex]\lambda_n x[/itex] convergent we cannot conclude the limit is a scalar multiple of [itex]x[/itex].
 
  • #4
Hi again!

As I look at the proof once again (assuming the proof is correct), the condition X to be a normed space could actually be reduced. The same proof will also be true for any metric space (X,d) with a metric d , as the metric induces the same metric on the subspaces.
 

1. What is a subspace in a normed 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.

2. How can we determine if a subset is a subspace of a normed vector space?

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.

3. What is the dimension of a subspace in a 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.

4. Can a subspace of a normed vector space be infinite-dimensional?

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.

5. What are some examples of subspaces in a normed vector space?

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.

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