Is There an Infinite-Dimensional Real Banach Space with a Non-Continuous Linear Functional?

Yes, there exists an infinite-dimensional real Banach space and a linear functional that is not continuous. This article will explore an example of such a space and functional, along with an explanation of the concepts involved.

Introduction to the Banach Space and Linear Functionals

A Banach space is a complete normed vector space. In an infinite-dimensional Banach space, it is possible to define a linear functional that is not continuous. Let's delve into an example to illustrate this.

Example: The Space of Absolutely Summable Sequences

Consider the Banach space (ell^1), the space of absolutely summable sequences. This space is defined as follows:

[ell^1 {x (x_1, x_2, x_3, ldots) : sum_{n1}^{infty} |x_n| The dual space of (ell^1) is (ell^{infty}), the space of bounded sequences:

[ell^{infty} {y (y_1, y_2, y_3, ldots) : sup_{n} |y_n| Define a linear functional [l: ell^1 to mathbb{R}] by:

[l(x) sum_{n1}^{infty} x_n]

This functional is continuous on (ell^1) because it is bounded. However, we can extend this to a larger space and construct a non-continuous functional.

Non-Continuous Functional Example: A Larger Space

Consider the space (ell^1 oplus mathbb{R}), which consists of pairs ((x, r)) where (x in ell^1) and (r in mathbb{R}). Define a functional (l: ell^1 oplus mathbb{R} to mathbb{R}) by:

[l(x, r) sum_{n1}^{infty} x_n r]

This functional is continuous.

Constructing a Non-Continuous Functional on an Infinite-Dimensional Space

To find a non-continuous functional, consider the space of all real sequences (mathbb{R}^{infty}), which is not complete. Define a functional (l: mathbb{R}^{infty} to mathbb{R}) by:

[l(x) begin{cases} 1 text{if } x_n 0 text{ for all but finitely many } n 0 text{otherwise} end{cases}]

This functional is linear but not continuous. To see why it is not continuous, consider a sequence (x^{k} (1, 1, ldots, 1, 0, 0, ldots)) where the first (k) entries are 1 and the rest are 0. The sequence (x^{k}) converges to the zero sequence in the standard topology for sequences, but:

[l(x^{k}) 1 forall k]and (l(0) 0). Therefore, the sequence (l(x^{k})) does not converge to (l(0)).

Conclusion

We have shown that in an infinite-dimensional real Banach space, it is possible to construct a linear functional that is not continuous. This demonstrates that such spaces and functionals indeed exist.

Continuous Linear Functionals

A continuous linear functional on a normed vector space is equivalent to a bounded linear functional. That is, a linear functional is continuous if and only if it is bounded in a neighborhood of 0.

For example, the spaces (L^p[a, b]) for (0 This article has provided a comprehensive explanation of the existence of non-continuous linear functionals in infinite-dimensional Banach spaces using rigorous mathematical concepts and examples.