Intereting Posts

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Atoms in a tail $\sigma$-algebra as $\liminf C_n$
primes represented integrally by a homogeneous cubic form
Text recommendation for introduction to linear algebra
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Combinatorially prove that $\sum_{i=0}^n {n \choose i} 2^i = 3^n $
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Assuming: $\forall x \in :f(x) > x$ Prove: $\forall x \in :f(x) > x + \varepsilon $
Is there always a prime between a prime and prime plus the index of that prime?
Image of the Riemann-sphere
Evaluate the integral $\int_{0}^{\infty} \frac{1}{(1+x^2)\cosh{(ax)}}dx$
Find the sum of $\sum 1/(k^2 – a^2)$ when $0<a<1$
Number of unit squares that meet a given diagonal line segment in more than one point

For a Banach space $V$ over $\mathbb{C}$ and $U \subset \mathbb{C}$ open, one can easily check that the notions of holomorphy hold for maps $f: U \rightarrow V$ just as in the classical sense. Indeed, I believe one can even use the Lebesgue integral for contour integration since Lebesgue integration on Banach-space-valued functions is also well-developed.

It seems that almost all the major results of classical complex analysis for holomorphic functions $f: U \rightarrow \mathbb{C}$ still hold in an analogous manner. Where are some crucial points where the theory differs when $f$ takes values in a Banach space?

- Counterexample of polynomials in infinite dimensional Banach spaces
- Trace operators on topological vector spaces
- Space of bounded functions is reflexive if the domain is finite
- Every closed subspace of ${\scr C}^0$ of continuously differentiable funcions must have finite dimension.
- Compact operator maps weakly convergent sequences into strongly convergent sequences
- An operator has closed range if and only if the image of some closed subspace of finite codimension is closed.

- Is there any connection between Green's Theorem and the Cauchy-Riemann equations?
- Why a holomorphic function satisfying these conditions has to be linear?
- Entire function bounded by polynomial of degree 3/2 must be linear.
- Integral $\int_0^\infty \exp(ia/x^2+ibx^2)dx$
- Fundamental theorem of calculus for complex analysis, proof
- Line contour integral of complex Gaussian
- Banach Spaces - How can $B,B',B'', B''', B'''',B''''',\ldots$ behave?
- Characterization of Harmonic Functions on the Punctured Disk
- If there is a branch of $\sqrt{z}$ on an open set $U$ with $0 \notin U,$ then there is also a branch of $arg$ $z.$
- Improper integrals with singularities on the REAL AXIS (Complex Variable)

L. Schwartz and A. Grothendieck made clear, by very early 1950s, that the Cauchy (-Goursat) theory of holomorphic functions of a single complex variable extended with essentially no change to functions with values in a locally convex, quasi-complete topological vector space. Cauchy integral formulas, residues, Laurent expansions, etc., all succeed (with trivial modifications occasionally).

Conceivably one needs a little care about the notion of “integral”. The Gelfand-Pettis “weak” integral suffices, but/and a Bochner version of “strong” integral is also available.

Further, in great generality, as Grothendieck made clear, “weak holomorphy” (that is, $\lambda\circ f$ holomorphic for all (continuous) linear functionals $\lambda$ on the TVS) implies (“strong”) holomorphy (i.e., of the TVS-valued $f$).

(Several aspects of this, and supporting matter, are on-line at http://www.math.umn.edu/~garrett/m/fun/Notes/09_vv_holo.pdf and other notes nearby on http://www.math.umn.edu/~garrett/m/fun/)

Edit: in response to @Christopher A. Wong’s further question… I’ve not made much of a survey of *recent* texts to see whether holomorphic TVS-valued functions are much discussed, but I would suspect that the main mention occurs in the setting of resolvents of operators on Hilbert and Banach spaces, abstracted just a little in abstract discussions of $C^*$ algebras. (Rudin’s “Functional Analysis” mentions weak integrals and weak/strong holomorphy and then doesn’t use them much, for example.) Schwartz’ original book did treat such things, and was the implied context for the first volume of the Gelfand-Graev-etalia “Generalized Functions”. In the latter, the examples are very small and tangible, but (to my taste) tremendously illuminating about *families* of distributions.

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