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

• D={z : ∣z∣<1}  • H (A→B) denotes the set of holomorfic  functions from A to B  • We define:  W={f∈H (D→R) : ∣∣f∣∣_W <∞ }  where  ∣∣ ∙ ∣∣_W  :  { (W,→,R_+ ),(f, ,(Σ_(n=0) ^∞ ((∣f^((n)) (0)∣)/(n!)))) :}    Let f∈W  Show that ∀g∈H ( f(D^� )), g○f∈W  tip: show that   ∣∣h∣∣_W ≤cste × Sup_(z∈D) {∣h(z)∣+∣h′′(z)∣}  and that W is an algebra    then, re−wright f=f_1 +f_2  with  f_2 : z  Σ_(n=N) ^∞ ((f^((n)) (0))/(n!))z^n   with N great enough to make sure that  Σ_(n=0) ^∞ ((g^((n)) (0))/(n!))f_2 ^( n)  is well defined and converges  over W.      ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙

$$\bullet\:{D}=\left\{{z}\::\:\mid{z}\mid<\mathrm{1}\right\} \\ $$ $$\bullet\:\mathscr{H}\:\left({A}\rightarrow{B}\right)\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{holomorfic} \\ $$ $$\mathrm{functions}\:\mathrm{from}\:{A}\:\mathrm{to}\:{B} \\ $$ $$\bullet\:\mathrm{We}\:\mathrm{define}: \\ $$ $${W}=\left\{{f}\in\mathscr{H}\:\left({D}\rightarrow\mathbb{R}\right)\::\:\mid\mid{f}\mid\mid_{{W}} <\infty\:\right\} \\ $$ $$\mathrm{where}\:\:\mid\mid\:\centerdot\:\mid\mid_{{W}} \::\:\begin{cases}{{W}}&{\rightarrow}&{\mathbb{R}_{+} }\\{{f}}&{ }&{\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mid{f}^{\left({n}\right)} \left(\mathrm{0}\right)\mid}{{n}!}}\end{cases} \\ $$ $$ \\ $$ $$\mathrm{Let}\:{f}\in{W} \\ $$ $$\mathrm{Show}\:\mathrm{that}\:\forall{g}\in\mathscr{H}\:\left(\:{f}\left(\bar {{D}}\right)\right),\:{g}\circ{f}\in{W} \\ $$ $${tip}:\:{show}\:{that} \\ $$ $$\:\mid\mid{h}\mid\mid_{{W}} \leqslant{cste}\:×\:\mathrm{Sup}_{{z}\in{D}} \left\{\mid{h}\left({z}\right)\mid+\mid{h}''\left({z}\right)\mid\right\} \\ $$ $${and}\:{that}\:{W}\:{is}\:{an}\:{algebra} \\ $$ $$ \\ $$ $$\mathrm{then},\:\mathrm{re}−\mathrm{wright}\:{f}={f}_{\mathrm{1}} +{f}_{\mathrm{2}} \:\mathrm{with} \\ $$ $${f}_{\mathrm{2}} :\:{z}\: \underset{{n}={N}} {\overset{\infty} {\sum}}\frac{{f}^{\left({n}\right)} \left(\mathrm{0}\right)}{{n}!}{z}^{{n}} \\ $$ $$\mathrm{with}\:{N}\:\mathrm{great}\:\mathrm{enough}\:\mathrm{to}\:\mathrm{make}\:\mathrm{sure}\:\mathrm{that} \\ $$ $$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{g}^{\left({n}\right)} \left(\mathrm{0}\right)}{{n}!}{f}_{\mathrm{2}} ^{\:{n}} \:\mathrm{is}\:\mathrm{well}\:\mathrm{defined}\:\mathrm{and}\:\mathrm{converges} \\ $$ $$\mathrm{over}\:{W}. \\ $$ $$\:\:\:\:\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot \\ $$

Commented byJDamian last updated on 12/Oct/22

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

Commented byTheHoneyCat 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 byTheHoneyCat last updated on 12/Oct/22

sorry if it wasn't obvious from the sentence.

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