Search found 4 matches

by Gigaster
Thu Jul 14, 2016 7:20 pm
Forum: Functional Analysis
Topic: Functional analysis - Exercises
Replies: 2
Views: 3571

Re: Functional analysis - Exercises

Hello!Here is a solution for the second excercise: Suppose \(T:X\rightarrow Y\) is bounded and \(f\in Y^{*}\).It follows that \(T\) is continuous on \(X\),so considering \(\{x_n,x\}\subset X\) such that \(x_n\rightarrow x \Rightarrow Tx_{n}\rightarrow Tx\).Since f is also continuous : \(f(Tx_n)\righ...
by Gigaster
Sun May 29, 2016 10:29 pm
Forum: General Topology
Topic: Compact
Replies: 1
Views: 2302

Re: Compact

Hello! Let \(A=\{x_n,x\}\subset X\) and \(\{V_i\}_{i\in I}\) an open cover of \(A\),i.e. \(V_i\in\mathbb T\quad\forall i\in I\) and \(A\subset\bigcup_{i\in I}V_i\). It follows that \(\forall n\in\mathbb N\quad\exists i_n\in I\) such that \(x_n\in V_{i_n}\) and also \(\exists i_x\in I\) such that \(x...
by Gigaster
Sun May 29, 2016 10:04 pm
Forum: Algebraic Structures
Topic: Exercise on ring theory
Replies: 2
Views: 2681

Re: Exercise on ring theory

Hello! Here is an answer to the second part.Let \(R\) be the ring of the 2X2 upper triangular real matrices (which is a subring of \(M_2(\mathbb R))\). Consider a mapping \( \phi : R\rightarrow\mathbb R \) such that \( \phi\left( \begin{array}{cc}a & b \\0 & c \end{array} \right)=a\) Obvious...
by Gigaster
Sat Jan 30, 2016 8:44 am
Forum: Functional Analysis
Topic: Hilbert space and sequence
Replies: 2
Views: 3092

Re: Hilbert space and sequence

Hello!Let \(a_{k}=\sum_{n=0}^{k} x_{n}\) and \(l\geq m\in\mathbb N\).Since \(x_{n}\) are mutually orthogonal,applying Pythagoras' theorem gives us that: $$ \|a_{l}-a_{m}\|^{2}=\|\sum_{n=m+1}^{l} x_{n}\|^{2}=\sum_{n=m+1}^{l}\|x_{n}\|^2=\sum_{n=1}^{l}\|x_{n}\|^2-\sum_{n=1}^{m}\|x_{n}\|^2 $$ It follows...