# L-infinity

**L-infinity** (written \( L^{\infty} \) or \( L^{\infty}(X,\mu) \)) is the [Banach space](https://www.edgepedia.org/banach-space) of essentially bounded measurable functions on a measure space, equipped with the essential supremum norm; its sequence-space counterpart \( \ell^{\infty} \) is the space of bounded sequences with the supremum norm. It is the endpoint \( p = \infty \) of the family of \( L^p \) spaces introduced by [Frigyes Riesz](https://www.edgepedia.org/frigyes-riesz) in 1910, and it measures a function by its largest value, ignoring what happens on sets of measure zero.<sup>[1](https://www.sciencedirect.com/topics/mathematics/abstract-space)</sup>

| Key fact | Detail |
|---|---|
| Elements | Essentially bounded Lebesgue measurable functions \( f : \Omega \to \mathbb{R} \) or \( \mathbb{C} \), meaning \( f \) is bounded on a subset whose complement has measure zero<sup>[2](https://www.math.ucdavis.edu/~hunter/book/ch5.pdf)</sup> |
| Norm | \( \|f\|_{\infty} = \inf\{c : \mu\{x : |f(x)| > c\} = 0\} \), the essential supremum<sup>[3](https://people.iith.ac.in/suku/operatortheory/node3.html)</sup> |
| Completeness | \( L^{\infty}(\Omega) \) is a Banach space for \( 1 \le p \le \infty \) across the \( L^p \) family<sup>[2](https://www.math.ucdavis.edu/~hunter/book/ch5.pdf)</sup> |
| Inclusions | On a space of positive finite measure, \( L^{\infty} \subsetneq L^p \subsetneq L^q \subsetneq L^1 \) for \( 1 < p \le q < \infty \)<sup>[3](https://people.iith.ac.in/suku/operatortheory/node3.html)</sup> |
| Algebra | \( L^{\infty}(S) \) is a unital commutative normed algebra under pointwise multiplication, with \( \|fg\|_{\infty} \le \|f\|_{\infty}\|g\|_{\infty} \)<sup>[4](https://web.pdx.edu/~erdman/CRA/COMPANION_ANALYSIS_pdf.pdf)</sup> |
| Dual | \( L^{\infty} \) is not reflexive except when the space has finitely many points; \( (\ell^{\infty})' \) is identified with the bounded finitely additive measures on \( \mathbb{N} \), strictly larger than \( \ell^1 \)<sup>[5](https://www.math.kit.edu/iana1/lehre/funcana2012w/media/fa-lecturenotes.pdf)</sup><sup> • </sup><sup>[6](https://repository.tudelft.nl/record/uuid%3A6b7baa94-665d-4b19-8797-e70d87ca7f8d)</sup> |
| Applied form | The \( H_{\infty} \) norm of a transfer function is the peak gain of its frequency response and the worst-case \( L_2 \) gain in the time domain<sup>[7](https://link.springer.com/article/10.1007/s40313-026-01260-2)</sup> |

## Definition and the essential supremum

Let \( (X, \mathcal{M}, \mu) \) be a measure space, meaning \( \mathcal{M} \) is a \( \sigma \)-algebra of subsets of \( X \) and \( \mu \) is a countably additive set-function with \( \mu(\varnothing) = 0 \).<sup>[8](https://www.math.unipd.it/~marson/didattica/Analisi_Funzionale/Dispensa_DeMarco.pdf)</sup> A function \( f \) is essentially bounded when it is bounded on a subset of its domain whose complement has measure zero, and its norm is the essential supremum<sup>[2](https://www.math.ucdavis.edu/~hunter/book/ch5.pdf)</sup>

\[ \|f\|_{\infty} = \inf\{c : \mu\{x : |f(x)| > c\} = 0\}. \]

The essential supremum is the maximum value of \( |f| \) if one is allowed to throw away a set of measure 0.<sup>[9](https://behemoth.cl.cam.ac.uk/dist_2021/HOL/HOL-Probability/Essential_Supremum.html)</sup> This differs from the ordinary pointwise supremum used for all bounded functions: \( L^{\infty} \) ignores changes on measure-zero sets. A function equal to 1 everywhere except on a null set, where it takes the value 1000, has \( \|f\|_{\infty} = 1 \).

Functions that differ only on a set of measure zero are identified, so an element of \( L^{\infty} \) is an equivalence class rather than a pointwise-defined function. Since a single point in a reasonable measure space is typically a null set, elements of \( L^p \) spaces do not generally have well-defined values at individual points; they have well-defined averages over measurable subsets.<sup>[10](https://www.mathematik.hu-berlin.de/~wendl/Winter2025/FunkAna/lecturenotes.pdf)</sup> With this identification the essential supremum is a genuine norm, and \( L^{\infty}(\mu) \), the space of essentially bounded functions, is complete in the metric it defines, making it a Banach space.<sup>[8](https://www.math.unipd.it/~marson/didattica/Analisi_Funzionale/Dispensa_DeMarco.pdf)</sup><sup> • </sup><sup>[11](https://planetmath.org/linftyxmu)</sup>

## Place among the Lp spaces

For \( 1 \le p < \infty \), \( L^p \) collects functions whose \( p \)-th power is integrable, with norm \( \|f\|_p = \left( \int |f|^p \, d\mu \right)^{1/p} \). The case \( p = \infty \) extends the family by replacing an integral condition with a boundedness condition.<sup>[3](https://people.iith.ac.in/suku/operatortheory/node3.html)</sup><sup> • </sup><sup>[2](https://www.math.ucdavis.edu/~hunter/book/ch5.pdf)</sup> The \( L^p \) spaces are Banach spaces for \( 1 \le p \le \infty \), and \( L^2 \) is the one member that is a [Hilbert space](https://www.edgepedia.org/hilbert-space).<sup>[2](https://www.math.ucdavis.edu/~hunter/book/ch5.pdf)</sup><sup> • </sup><sup>[3](https://people.iith.ac.in/suku/operatortheory/node3.html)</sup>

**Inclusions reverse with p.** On a measure space of positive finite measure, boundedness is the strongest condition: for \( 1 < p \le q < \infty \),

\[ L^{\infty} \subsetneq L^p \subsetneq L^q \subsetneq L^1, \]

with the norm inequality \( \|f\|_q \le \mu(X)^{(1/p - 1/q)} \|f\|_p \) connecting the scales.<sup>[3](https://people.iith.ac.in/suku/operatortheory/node3.html)</sup>

## The dual space

For \( 1 \le p < \infty \), the dual of \( L^p \) is \( L^q \) where \( 1/p + 1/q = 1 \), a duality Riesz established between 1910 and 1916.<sup>[12](https://link.springer.com/article/10.1007/s00283-026-10563-w)</sup> The endpoint \( p = \infty \) breaks the pattern. \( L^{\infty} \) is the dual of \( L^1 \), but it is not reflexive except when the underlying space \( \Omega \) consists of a finite number of points; consequently \( (L^{\infty})^* \) is strictly bigger than \( L^1 \), and there are continuous linear functionals on \( L^{\infty} \) that cannot be written as \( \varphi(f) = \int u f \, d\mu \) for any \( u \in L^1 \).<sup>[5](https://www.math.kit.edu/iana1/lehre/funcana2012w/media/fa-lecturenotes.pdf)</sup>

For the sequence space, a 2023 TU Delft bachelor thesis by R. Schermerhorn, mentored by K. P. Hart, proves that the natural embedding of \( \ell^1 \) into \( (\ell^{\infty})' \) fails to be surjective and that no isometric isomorphism, or even a homeomorphism, between the two spaces exists.<sup>[6](https://repository.tudelft.nl/record/uuid%3A6b7baa94-665d-4b19-8797-e70d87ca7f8d)</sup> The thesis identifies \( (\ell^{\infty})' \) with the space \( \mathrm{ba}(\mathbb{N}, \mathcal{P}(\mathbb{N})) \) of bounded finitely additive measures on \( \mathbb{N} \), a class that includes measures induced by \( \ell^1 \), diffuse measures, shift-invariant measures, and 0,1-valued measures, the last corresponding to ultrafilters on \( \mathbb{N} \) and to multiplicative functionals on \( \ell^{\infty} \).<sup>[6](https://repository.tudelft.nl/record/uuid%3A6b7baa94-665d-4b19-8797-e70d87ca7f8d)</sup>

\( L^{\infty} \) is also large in a topological sense: \( L^{\infty}(\mathbb{R}^d) \) is not separable, and \( L^{\infty}(\Omega) \) fails separability except when \( \Omega \) consists of finitely many atoms.<sup>[5](https://www.math.kit.edu/iana1/lehre/funcana2012w/media/fa-lecturenotes.pdf)</sup> Bounded sequences in \( L^{\infty} \) do admit weak*-convergent subsequences, a consequence of \( L^{\infty} \) being the dual of the separable space \( L^1 \).<sup>[5](https://www.math.kit.edu/iana1/lehre/funcana2012w/media/fa-lecturenotes.pdf)</sup>

## Algebra structure

\( L^{\infty} \) carries more structure than a normed vector space. Because the product of two essentially bounded functions is essentially bounded and \( \|fg\|_{\infty} \le \|f\|_{\infty} \|g\|_{\infty} \), pointwise multiplication is continuous, and \( L^{\infty}(S) \) is a unital commutative normed algebra, with the constant function 1 as its unity.<sup>[4](https://web.pdx.edu/~erdman/CRA/COMPANION_ANALYSIS_pdf.pdf)</sup><sup> • </sup><sup>[13](https://math.i-learn.unito.it/pluginfile.php/241932/mod_resource/content/34/FunctionalAnalysis.pdf)</sup> For a nondiscrete \( \sigma \)-compact locally compact Hausdorff group \( G \), \( L_{\infty}(G) \) under pointwise multiplication has many nonzero proper closed invariant ideals, including a continuum of maximal invariant ideals.<sup>[14](https://www.sciencedirect.com/science/article/pii/0022123676900276)</sup>

Within \( \ell^{\infty}(\Lambda) \) sits the important subspace \( c_0(\Lambda) \) of functions that are 0 at infinity: \( x \) belongs to \( c_0 \) when for every \( \varepsilon > 0 \) the set \( \{\lambda : |x(\lambda)| > \varepsilon \} \) is finite.<sup>[8](https://www.math.unipd.it/~marson/didattica/Analisi_Funzionale/Dispensa_DeMarco.pdf)</sup>

## Applications

The supremum norm is the working norm wherever a worst-case bound is the quantity of interest. In robust control, the \( H_{\infty} \) norm of a closed-loop transfer function, defined as the supremum of the largest singular value over frequencies, represents the peak gain of the system's [Bode plot](https://www.edgepedia.org/bode-plot) and, in the time domain, the worst-case \( L_2 \) gain; \( H_{\infty} \) synthesis minimizes this norm to guarantee protection against worst-case perturbations.<sup>[7](https://link.springer.com/article/10.1007/s40313-026-01260-2)</sup><sup> • </sup><sup>[15](https://link.springer.com/chapter/10.1007/978-3-319-91707-8_5)</sup> For discrete-time systems the \( \ell^1 \) norm determines peak gain while the \( H_{\infty} \) norm determines RMS gain, and the two can differ substantially: a convolution system can have a frequency response small at all frequencies yet greatly amplify the peaks of signals passing through it, though for finite-dimensional systems of dimension \( n \) the peak gain exceeds the RMS gain by at most a factor \( 2n + 1 \).<sup>[16](https://stanford.edu/~boyd/papers/ell_one_optimal.html)</sup><sup> • </sup><sup>[17](https://web.stanford.edu/~boyd/papers/peak_RMS_gain.html)</sup> The \( L_{\infty} \) norm also appears directly in engineering software as the peak gain of a frequency response across frequencies.<sup>[18](https://www.mathworks.com/help/control/ref/dynamicsystem.norm.html)</sup>

In approximation theory, minimization in the \( l^{\infty} \) norm is [Chebyshev approximation](https://www.edgepedia.org/chebyshev-approximation), the minimax problem of minimizing the largest error, with a literature connecting \( L^p \) approximations to Chebyshev ones.<sup>[19](https://epubs.siam.org/doi/10.1137/0111074)</sup>

## References

1. [Abstract Space, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/abstract-space)
2. [Banach Spaces, Chapter 5, J. Hunter, UC Davis](https://www.math.ucdavis.edu/~hunter/book/ch5.pdf)
3. [Infinite dimensional vector space, Operator Theory notes, IIT Hyderabad](https://people.iith.ac.in/suku/operatortheory/node3.html)
4. [Companion to Real Analysis, E. Erdman, Portland State University](https://web.pdx.edu/~erdman/CRA/COMPANION_ANALYSIS_pdf.pdf)
5. [Functional Analysis lecture notes, Karlsruhe Institute of Technology](https://www.math.kit.edu/iana1/lehre/funcana2012w/media/fa-lecturenotes.pdf)
6. [The dual space of L-infinity, R. Schermerhorn, TU Delft Repository (2023)](https://repository.tudelft.nl/record/uuid%3A6b7baa94-665d-4b19-8797-e70d87ca7f8d)
7. [H∞ Robust Sampled-Data State-Feedback Control of Linear Systems, Journal of Control, Automation and Electrical Systems (2026)](https://link.springer.com/article/10.1007/s40313-026-01260-2)
8. [Banach spaces, G. DeMarco, Università di Padova](https://www.math.unipd.it/~marson/didattica/Analisi_Funzionale/Dispensa_DeMarco.pdf)
9. [Theory Essential_Supremum, Isabelle/HOL Probability library](https://behemoth.cl.cam.ac.uk/dist_2021/HOL/HOL-Probability/Essential_Supremum.html)
10. [Functional Analysis lecture notes, Humboldt University Berlin](https://www.mathematik.hu-berlin.de/~wendl/Winter2025/FunkAna/lecturenotes.pdf)
11. [L^∞(X,μ), PlanetMath](https://planetmath.org/linftyxmu)
12. [From Riesz to Kakutani: Revealing Foundations of Probability via Representation Theorems, The Mathematical Intelligencer](https://link.springer.com/article/10.1007/s00283-026-10563-w)
13. [Functional Analysis lecture notes, Università di Torino](https://math.i-learn.unito.it/pluginfile.php/241932/mod_resource/content/34/FunctionalAnalysis.pdf)
14. [Invariant means and invariant ideals in L∞(G), Journal of Functional Analysis](https://www.sciencedirect.com/science/article/pii/0022123676900276)
15. [Performance and Robustness Measures, S. A. Frank, Control Theory Tutorial, Springer (2018)](https://link.springer.com/chapter/10.1007/978-3-319-91707-8_5)
16. [Design of l1 Optimal Controllers, J. B. Pearson and S. Boyd (1987)](https://stanford.edu/~boyd/papers/ell_one_optimal.html)
17. [Comparison of Peak and RMS Gains for Discrete-Time Systems, S. Boyd and J. Doyle, Systems and Control Letters (1987)](https://web.stanford.edu/~boyd/papers/peak_RMS_gain.html)
18. [norm - Norm of linear model, MATLAB documentation](https://www.mathworks.com/help/control/ref/dynamicsystem.norm.html)
19. [Approximations in Lp and Chebyshev Approximations, SIAM Journal on Applied Mathematics](https://epubs.siam.org/doi/10.1137/0111074)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Operator algebras › Banach and normed algebras › Examples and special classes of Banach algebras*

*Initially written Sep 17, 2026 · Reviewed: Sep 19, 2026; Sep 30, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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