Understanding Inner Product Spaces and Norms in Infinite-Dimensional Vector Spaces

When dealing with infinite-dimensional normed vector spaces, it becomes crucial to understand the differences and relationships between these spaces and inner product spaces. This article will explore whether every infinite-dimensional normed vector space is also an inner product Hilbert space, and discuss the necessary conditions and counterexamples.

The Role of Inner Product Spaces in Normed Vector Spaces

Every inner product space has an associated norm given by v sqrt{langle vv rangle}. This norm is derived from the inner product, and it satisfies certain conditions, including the Parallelogram Law. However, not all norms are derived from inner products. This raises the question of whether every normed vector space can be considered an inner product space.

Counterexample: The Space of Bounded Sequences with Supremum Norm

One well-known counterexample is the space of all bounded sequences, denoted by m. This space is equipped with the supremum norm, defined as x sup xi. The supremum norm does not satisfy the Parallelogram Law, which means it cannot be derived from an inner product. Therefore, not every norm comes from an inner product.

Completeness and Inner Product Spaces

Furthermore, not every space with an inner product is complete under the corresponding norm. Completeness is a necessary condition for a space to be an inner product space, but it is not sufficient. Additionally, completeness plays a crucial role in defining the properties of the dual space. In finite dimensions, completeness is automatic, but this is not the case in infinite dimensions.

Norms and Completeness in Infinite-Dimensional Spaces

There are norms that can be induced by inner products, but not all norms are induced by inner products. A norm is induced by an inner product if and only if the norm squared of a sum less the sum of the norms is bilinear. This condition is known as the Parallelogram Law.

Dual Spaces and Completeness

Completeness is a key concept in the theory of inner product spaces. The dual of the dual of a complete inner product space is the original space, which is not generally true for complete normed spaces. This highlights the importance of completeness in the structure of these spaces.

Finite-Dimensional vs. Infinite-Dimensional Spaces

In finite-dimensional real or complex vector spaces, every vector space is isomorphic to a finite product of the field with itself, and the usual sum of squares of absolute values is the norm that comes from the inner product. To go from a norm to a bilinear form, one can compare the squares of the norms of sums with the sums of the squares of the norms.

Finite Fields and Inner Products

Finally, it's important to note that vector spaces defined over finite fields cannot be equipped with a well-defined inner product due to the lack of an order on finite fields. This implies that finite fields cannot satisfy the positive-definite property required for well-defined inner products.

Conclusion

While every finite-dimensional normed vector space can be considered an inner product space, this is not the case in infinite dimensions. The Parallelogram Law and completeness are crucial conditions that must be satisfied for a norm to come from an inner product. Understanding these concepts is essential for anyone working with infinite-dimensional normed vector spaces and their applications in various fields of mathematics and science.