Prove that an idempotent linear operator of a Hilbert space is self-adjoint if and only if it has norm $ 0$ or $ 1$. [i]J. Szucs[/i]

Let $ A$ be a family of proper closed subspaces of the Hilbert space $ H\equal{}l^2$ totally ordered with respect to inclusion (that is , if $ L_1,L_2 \in A$, then either $ L_1\subset L_2$ or $ L_2\subset L_1$). Prove that there exists a vector $ x \in H$ not contaied in any of the subspaces $ L$ belonging to $ A$. [i]B. Szokefalvi Nagy[/i]

Let be a sequence of functions $ \left( f_n \right)_{n\ge 2}:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ defined, for each $ n\ge 2, $ as $$ f_n(x)=2nx^{2+n} -2(n+2)x^{1+n} +(2+n)x +1. $$ [b]a)[/b] Prove that $ f_n $ has an unique local maxima $ x_n, $ for any $ n\ge 2. $ [b]b)[/b] Show that $ 1=\lim_{n\to\infty } x_n. $ [i]Cătălin Zîrnă[/i]

A function $f:\mathbb R\to\mathbb R$ has the property that for every $x,y\in\mathbb R$ there exists a real number $t$ (depending on $x$ and $y$) such that $0<t<1$ and $$f(tx+(1-t)y)=tf(x)+(1-t)f(y).$$ Does it imply that $$f\left(\frac{x+y}2\right)=\frac{f(x)+f(y)}2$$ for every $x,y\in\mathbb R$?

[b]7.[/b] Define the generalized derivative at $x_0$ of the function $f(x)$ by $\lim_{h \to 0} 2 \frac{ \frac{1}{h} \int_{x_0}^{x_0+h} f(t) dt - f(x_0)}{h}$ Show that there exists a function, continuous everywhere, which is nowhere differentiable in this general sense [b]( R. 8)[/b]

Let $0<a<1$. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ continuous at $x = 0$ such that $f(x) + f(ax) = x,\, \forall x \in \mathbb{R}$

Let $ A\subset \mathbb C$ be a closed and countable set. Prove that if the analytic function $ f: \mathbb C\backslash A\longrightarrow \mathbb C$ is bounded, then $ f$ is constant.

Let $H$ be the hyperbolic plane, $I(H)$ be the isometry group of $H$, and $O\in H$ be a fixed starting point. Determine those continuous $\sigma\colon H\rightarrow I(H)$ mappings that satisfty the following three conditions: (a) $\sigma(O)=\mathrm{id}$, and $\sigma (X)O=X$ for all $X\in H$; (b) for every $X\in H\backslash \{ O\}$ point, the $\sigma(X)$ isometry is a paracyclic shift, i.e. every member of a system of paracycles through a common infinitely far point is left invariant; (c) for any pair $P,Q\in H$ of points there exists a point $X\in H$ such that $\sigma(X)P=Q$. Prove that the $\sigma\colon H\rightarrow I(H)$ mappings satisfying the above conditions are differentiable with the exception of a point.

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuously differentiable,strictly convex function.Let $H$ be a Hilbert space and $A,B$ be bounded,self adjoint linear operators on $H$.Prove that,if $f(A)-f(B)=f'(B)(A-B)$ then $A=B$.

Let $H$ be a complex Hilbert space. The bounded linear operator $A$ is called [i]positive[/i] if $\langle Ax, x\rangle \geq 0$ for all $x\in H$. Let $\sqrt A$ be the positive square root of $A$, i.e. the uniquely determined positive operator satisfying $(\sqrt{A})^2=A$. On the set of positive operators we introduce the $$A\circ B=\sqrt A B\sqrt B$$ operation. Prove that for a given pair $A, B$ of positive operators the identity $$(A\circ B)\circ C=A\circ (B\circ C)$$ holds for all positive operator $C$ if and only if $AB=BA$.

Let $ \mathcal{H}$ be an infinite-dimensional Hilbert space, let $ d>0$, and suppose that $ S$ is a set of points (not necessarily countable) in $ \mathcal{H}$ such that the distance between any two distinct points in $ S$ is equal to $ d$. Show that there is a point $ y\in\mathcal{H}$ such that \[ \left\{\frac{\sqrt{2}}{d}(x\minus{}y): \ x\in S\right\}\] is an orthonormal system of vectors in $ \mathcal{H}$.

Let H be a complex separable Hilbert space and denote $B(H)$ the algebra of bounded linear operators on H. Find all *-subalgebras C of $B(H)$ for which for all $A \in B(H)$ and $T \in C$ there exists $S \in C$ that $$TA-AT^{\ast} = TS-ST^{\ast}$$ note: *-algebra is also known as involutive algebra.

Let $ I $ be a nondegenerate interval, and let $ F $ be a primitive of a function $ f:I\longrightarrow\mathbb{R} . $ Show that for any distinct $ a,b\in I, $ the tangents to the graph of $ F $ at the points $ (a,F(a)) ,(b,F(b)) $ are concurrent at a point whose abscisa is situated in the interval $ (a,b). $ [i]Nicolae Bourbăcuț[/i]

[b]8.[/b] Let $(a_n)_{n=1}^{\infty}$ be a sequence of positive numbers and suppose that $\sum_{n=1}^{\infty} a_n^2$ is divergent. Let further $0<\epsilon<\frac{1}{2}$. Show that there exists a sequence $(b_n)_{n=1}^{\infty}$ of positive numbers such that $\sum_{n=1}^{\infty}b_n^2$ is convergent and $\sum_{n=1}^{N}a_n b_n >(\sum_{n=1}^{N}a_n^2)^{\frac{1}{2}-\epsilon}$ for every positive integer $N$. [b](S. 8)[/b]

Let $C[-1,1]$ be the space of continuous real functions on the interval $[-1,1]$ with the usual supremum norm, and let $V{}$ be a closed, finite-codimensional subspace of $C[-1,1].$ Prove that there exists a polynomial $p\in V$ with norm at most one, which satisfies $p'(0)>2023.$

Let H be a Hilbert space over the field of real numbers $\Bbb R$. Find all $f: H \to \Bbb R$ continuous functions for which $$f(x + y + \pi z) + f(x + \sqrt{2} z) + f(y + \sqrt{2} z) + f (\pi z)$$ $$= f(x + y + \sqrt{2} z) + f (x + \pi z) + f (y + \pi z) + f(\sqrt{2} z)$$ is satisfied for any $x , y , z \in H$.

Let $\theta$ and $p(p<1)$ ) be nonnegative real numbers. Suppose that $f:X\to Y$ is mapping with $f(0)=0$ and $$\left |\left| 2f\left (\frac{x+y}{2}\right )-f(x)-f(y) \right |\right|_Y \leq \theta\left (\left |\left |x\right |\right |_X^p +\left |\left |y\right |\right |_X^p \right )$$ for all $x$, $y\in \mathbb{Z}$ with $x\perp y$ where $X$ is an orthogonality space and $Y$ is a real Banach space. Prove that there exists a unique orthogonally Jensen additive mapping $T:X\to Y$, namely a mapping $T$ that satisfies the so-called orthogonally Jensen additive functional equation $$2f\left (\frac{x+y}{2}\right )=f(x)+f(y)$$for all $x$, $y\in \mathbb{X}$ with $x\perp y$, satisfying the property $$\left |\left|f(x)-T(x) \right |\right|_Y \leq \frac{2^p\theta}{2-2^p}\left |\left |x\right |\right |_X^p$$ for all $x\in X$

Let $n\ge 1$ be a fixed integer. Calculate the distance $\inf_{p,f}\, \max_{0\le x\le 1} |f(x)-p(x)|$ , where $p$ runs over polynomials of degree less than $n$ with real coefficients and $f$ runs over functions $f(x)= \sum_{k=n}^{\infty} c_k x^k$ defined on the closed interval $[0,1]$ , where $c_k \ge 0$ and $\sum_{k=n}^{\infty} c_k=1$.

Let $N$ be a normed linear space with a dense linear subspace $M$. Prove that if $L_1,\ldots,L_m$ are continuous linear functionals on $N$, then for all $x\in N$ there exists a sequence $(y_n)$ in $M$ converging to $x$ satisfying $L_j(y_n)=L_j(x)$ for all $j=1,\ldots,m$ and $n\in \mathbb{N}$.

Let $ T$ be a bounded linear operator on a Hilbert space $ H$, and assume that $ \|T^n \| \leq 1$ for some natural number $ n$. Prove the existence of an invertible linear operator $ A$ on $ H$ such that $ \| ATA^{\minus{}1} \| \leq 1$. [i]E. Druszt[/i]

Let H be an infinite dimensional, separable, complex Hilbert space and denote $\cal B (\cal H)$ the $\cal H$-algebra of its bounded linear operators. Consider the algebras $l_{\infty} ({\Bbb N}, \cal B (\cal H) ) = $ $\{ (a_n) | A_n \in\cal B (\cal H)$ $(n \in {\Bbb N}), \sup_n ||A_n|| <\infty \}$ $C(\beta {\Bbb N}, \cal B (\cal H) )$ = $\{ f: \beta {\Bbb N} \to \cal B (\cal H)|$ f is continuous $\}$ with pointwise operations and supremum norm. Show that these C*-algebras are not isometrically isomorphic. (Here, $\beta {\Bbb N}$ denotes the Stone-Cech compactification of the set of natural numbers.)