Higher Frechet derivatives and Taylor formula in normed spaces

Prove the stated assertion. A bilinear form $F(x ; y), x \in X, y \in X$, is called symmetric if for any elements $x \in X$ and $y \in X$ the equality $F(x ; y) = F(y ; x)$ holds. If a mapping $f: G \rightarrow Y$ of a set $G$ open in the space $X$ has a second derivative $f^{\prime \prime }$ at a point $x_{0} \in G$, then this derivative is a symmetric form belonging to the space $\mathscr {L}_{2}\left(X^{2} ; Y\right) \equiv \mathscr {L}_{2}(X, X ; Y)$ (see problem 159).

Prove the stated assertion. The formula

holds.

Prove the stated assertion. The formula

holds.

Prove the stated assertion. The derivative $f^{(n)}$ of any order $n$ is a symmetric $n$-linear form (an $n$-linear form $F\left(x_{1} ; \ldots ; x_{n}\right)$ is called symmetric if its values do not change under any permutation of its arguments).

Prove the stated assertion. If a mapping $f: G \rightarrow Y$ of a set $G$ open in the space $X$ has $n$ continuous derivatives on the segment $\left[x_{0} ; x\right] \subset G$ and a derivative of order $n+1$ on the interval $\left(x_{0} ; x\right)$, then

Prove the stated assertion. If, under the assumptions of the preceding problem, the derivative of order $n+1$ of the mapping $f$ is bounded on the interval $\left(x_{0} ; x\right)$:

then

$f\left(x_{0}+h\right) = f\left(x_{0}\right)+f^{\prime }\left(x_{0}\right) h+\frac{f^{\prime \prime }\left(x_{0}\right)}{2 !} h^{2}+\ldots +\frac{f^{(n)}\left(x_{0}\right)}{n !} h^{n}+o\left(h^{n}\right)$

$h \rightarrow 0$

(Taylor's formula).

For the mapping $f: C^{2}[0 ; 1] \rightarrow C[0 ; 1]$ (see problem 69) $(f(x))(t) = \frac{d^{2} x}{d^{2} t}+\sin x(t)$, find all its derivatives at the point $x = t$ and expand it by Taylor's formula in a neighborhood of this point.