Proving that Every Subspace of an Infinite Dimensional Vector Space is Still Infinite Dimensional

Introduction

In the realm of abstract mathematics, the concept of a vector space plays a pivotal role. A vector space can be finite or infinite dimensional, depending on its basis. The dimension of a vector space is the cardinality of its basis set. In this article, we delve into the properties of vector spaces, focusing on infinite dimensional spaces—a topic with profound implications in various fields of mathematics and physics.

This article discusses the assertion that "every subspace of an infinite dimensional vector space is also infinite dimensional." We provide a rigorous proof and examples to substantiate this statement, even in the case of non-normed real vector spaces. This discussion is particularly relevant in the context of understanding the behavior of subspaces in abstract algebra and functional analysis.

Definitions and Preliminaries

A vector space is a set of elements called vectors, which can be added together and multiplied (scaled) by scalars. The scalars typically come from a field, such as the real or complex numbers. When the scalars are real numbers, the vector space is called a real vector space.

The dimension of a vector space is the number of elements in a basis, which is a linearly independent set that spans the vector space. A vector space is called finite dimensional if it has a finite basis, and infinite dimensional if it has an infinite basis.

A subspace of a vector space is a subset that is also a vector space under the same operations. Every subspace of a finite dimensional vector space is itself finite dimensional. However, the question at hand is whether the same holds for infinite dimensional vector spaces, specifically in contexts where the space is not necessarily normed.

Proof that Every Subspace of an Infinite Dimensional Vector Space is Infinite Dimensional

Claim: In an infinite-dimensional vector space V over the real numbers, every non-trivial subspace W is infinite-dimensional.

Proof:

Consider an infinite-dimensional vector space V. By definition, V has an infinite basis ({e_i}_{i in I}), where (I) is an infinite index set. Assume (W) is a non-trivial subspace of (V). This means (W) is not the zero subspace ({0}) and contains at least one non-zero vector. We aim to show that (W) must also be infinite-dimensional.

Since (W) is non-trivial, pick any non-zero vector (x in W). Clearly, ({x}) is a one-dimensional subspace of (W). However, (W) itself might be infinite-dimensional. To show this, we need to demonstrate that any such (x) cannot span a finite basis for (W).

Consider that if (W) were finite-dimensional, say with dimension (n), then any basis for (W) would consist of (n) linearly independent vectors. But since (x) is a non-zero vector in (W), it can be part of a linearly independent set. If (W) were finite-dimensional, there would exist a finite set of vectors in (W) that could span (W). However, this contradicts the fact that (V) is infinite-dimensional. Therefore, (W) must also be infinite-dimensional.

Examples and Further Insights

For a concrete example, consider the vector space (mathbb{R}^mathbb{N}), the space of all real-valued sequences, which is infinite-dimensional. Let (W) be the subspace of (mathbb{R}^mathbb{N}) consisting of all sequences with only finitely many non-zero terms. It is clear that (W) is not the zero subspace, as it contains non-zero sequences. Moreover, (W) is an infinite-dimensional subspace because any finite linearly independent subset (e.g., the standard basis vectors ((1, 0, 0, dots), (0, 1, 0, dots),dots)) can be extended to an infinite basis set by including sequences with just one non-zero term in different positions.

In another scenario, consider the vector space (C(mathbb{R})) of all continuous functions from (mathbb{R}) to (mathbb{R}). This space is infinite-dimensional, and any non-trivial subspace, such as the subspace of all polynomials, is also infinite-dimensional. For instance, the set of polynomials ({1, x, x^2, dots}) is a basis for the space of all polynomials, showing that it is infinite-dimensional.

Conclusion

The key takeaway from this discussion is that in infinite-dimensional vector spaces, every non-trivial subspace retains the infinite-dimensional property, irrespective of whether the space is normed or not. This is a fundamental property that holds true across a broad spectrum of mathematical contexts, from linear algebra to functional analysis.

Understanding the behavior of subspaces in infinite-dimensional vector spaces is crucial for many applications, including in the development of advanced mathematical theories and practical problem-solving techniques in various fields. Whether one is working with normed or non-normed spaces, the cardinality of the basis set remains a key factor.

Key Concepts and Keywords

Keywords: Vector Space, Infinite Dimensional, Subspace, Non-Normed Space

References:

Rudin, W. (1991). . McGraw-Hill Education. Hoffmann, K. R., Kunze, R. (1971). . Prentice-Hall. Conway, J. B. (1990). . Springer.