Math 2.0

Functional analysis Definition

Analytic function between Banach spaces

Builds on Banach spaceMultilinear mapOperator normAnalytic functionSmooth non-analytic function

Definition

Let 𝕂\mathbb{K} be ℝ\R or ℂ\C, and let XX and YY be Banach spaces over 𝕂\mathbb{K}. For n≥1n \ge 1, let Ln(X,Y)L^n(X, Y) be the space of continuous multilinear maps A ⁣:Xn→YA \colon X^n \to Y, with the norm ∥A∥=sup⁡{∥A(h1,…,hn)∥:∥h1∥,…,∥hn∥≤1}\norm{A} = \sup \set{\norm{A(h_1, \ldots, h_n)} : \norm{h_1}, \ldots, \norm{h_n} \le 1}. Let L0(X,Y)=YL^0(X, Y) = Y. Let U⊆XU \subseteq X be open, let f ⁣:U→Yf \colon U \to Y and let a∈Ua \in U. The map ff is analytic at aa if there are An∈Ln(X,Y)A_n \in L^n(X, Y) for n≥0n \ge 0 and a real r>0r > 0 such that:

  1. sup⁡n∥An∥rn<∞\sup_n \norm{A_n} r^n < \infty.
  2. f(a+h)=∑n=0∞An(h,…,h)f(a + h) = \sum_{n=0}^{\infty} A_n(h, \ldots, h) for all h∈Xh \in X with ∥h∥<r\norm{h} < r.

The map ff is analytic on UU if it is analytic at each point of UU.

Remarks

Clause 1 makes the series in clause 2 converge absolutely, since ∥An(h,…,h)∥≤∥An∥∥h∥n\norm{A_n(h, \ldots, h)} \le \norm{A_n} \norm{h}^n. The maps h↦An(h,…,h)h \mapsto A_n(h, \ldots, h) are the continuous homogeneous polynomials of degree nn; one may take each AnA_n symmetric. For X=Y=ℝX = Y = \R, this is the usual analytic function. An analytic map is C∞C^\infty in the sense of the Fréchet derivative.

Example: let BB be a Banach algebra with unit 11 and ∥xy∥≤∥x∥∥y∥\norm{xy} \le \norm{x} \norm{y}, and let f(x)=(1−x)−1f(x) = (1 - x)^{-1} for ∥x∥<1\norm{x} < 1. Take An(h1,…,hn)=h1h2⋯hnA_n(h_1, \ldots, h_n) = h_1 h_2 \cdots h_n, so ∥An∥≤1\norm{A_n} \le 1 for n≥1n \ge 1, and r=1r = 1. The Neumann series ∑nxn\sum_n x^n converges and equals (1−x)−1(1 - x)^{-1}, so ff is analytic at 00.

Non-example: the smooth non-analytic function f(t)=e−1/t2f(t) = e^{-1/t^2}, f(0)=0f(0) = 0, on ℝ\R is not analytic at 00. All its derivatives at 00 vanish, so any series as in clause 2 would be 00, but f(t)≠0f(t) \neq 0 for t≠0t \neq 0.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

Show the path as a graph
Analytic function, concept, no evidence levelAnalytic functionconceptBanach space, concept, no evidence levelBanach spaceconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionMultilinear map, concept, no evidence levelMultilinear mapconceptOperator norm, concept, no evidence levelOperator normconceptSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionSeries, definition, no evidence levelSeriesdefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionPower series, definition, no evidence levelPower seriesdefinitionDerivative function, definition, no evidence levelDerivative functiondefinitionAnalytic function, definition, no evidence levelAnalytic functiondefinitionHigher derivative, definition, no evidence levelHigher derivativedefinitionTaylor polynomial, definition, no evidence levelTaylor polynomialdefinitionContinuously differentiable function, definition, no evidence levelContinuously differen…definitionSmooth function, definition, no evidence levelSmooth functiondefinitionTaylor series, definition, no evidence levelTaylor seriesdefinitionSmooth non-analytic function, counterexample, no evidence levelSmooth non-analytic f…counterexampleAnalytic function between Banach spaces, definition, no evidence levelAnalytic function bet…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Analytic function concept
  2. Banach space concept
  3. Limit point of a set definition
  4. Multilinear map concept
  5. Operator norm concept
  6. Sequence definition
  7. Limit of a function definition
  8. Real sequence definition
  9. Continuity at a point definition
  10. Derivative definition
  11. Series definition
  12. Continuous function definition
  13. Differentiable function definition
  14. Power series definition
  15. Derivative function definition
  16. Analytic function definition
  17. Higher derivative definition
  18. Taylor polynomial definition
  19. Continuously differentiable function definition
  20. Smooth function definition
  21. Taylor series definition
  22. Smooth non-analytic function counterexample
  23. Analytic function between Banach spaces definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Analytic_function
About this page
TagB538T38WDS
URIm2://B538T38WDS
Kinddefinition
WritingW0 stub
MSC46G20, 26E05
DefinesAnalytic function between Banach spaces
UsesBanach space, Multilinear map, Operator norm, Analytic function, Smooth non-analytic function
Changed2 November 2026

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