D-z-z-lt-1-H-A-B-denotes-the-set-of-holomorfic-functions-from-A-to-B-We-define-W-f-H-D-R-f-W-lt-where-W-W-R-f-n-0-f-n-0-n-

Question Number 178032 by TheHoneyCat last updated on 12/Oct/22

∙ D = {z : ∣ z ∣< 1}

∙ H (A \to B) denotes the set of holomorfic

functions from A to B

∙ We define :

W = {f \in H (D \to R) : ∣∣ f ∣ ∣_{W} < \infty}

where ∣∣ \cdot ∣ ∣_{W} : {\begin{cases} W & \to & R_{+} \\ f & \underset{n = 0}{\sum^{\infty}} \frac{∣ f^{(n)} (0) ∣}{n!} \end{cases}

Let f \in W

Show that \forall g \in H (f (\bar{D})), g \circ f \in W

t i p : s h o w t h a t

∣∣ h ∣ ∣_{W} ⩽ c s t e \times {Sup}_{z \in D} {∣ h (z) ∣ + ∣ h^{″} (z) ∣}

a n d t h a t W i s a n a l g e b r a

then, re - wright f = f_{1} + f_{2} with

f_{2} : z \underset{n = N}{\sum^{\infty}} \frac{f^{(n)} (0)}{n!} z^{n}

with N great enough to make sure that

\underset{n = 0}{\sum^{\infty}} \frac{g^{(n)} (0)}{n!} f_{2}^{n} is well defined and converges

over W .

\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot

Commented by JDamian last updated on 12/Oct/22

Besides I do not understand a half of your question, what damn is h? And cste?

Commented by TheHoneyCat last updated on 12/Oct/22

"h" is any function of W (so that you can take its norm). "cste" is any constant (so that it can be multiplied).

Commented by TheHoneyCat last updated on 12/Oct/22

sorry if it wasn't obvious from the sentence.