\documentclass{article} \usepackage{graphicx} % Required for inserting images \title{6.3.11} \author{Joshua Hoover} \date{September 2025} \begin{document} \maketitle \begin{equation} 0=⟨T^*Tv,v⟩=⟨Tv,Tv⟩ \end{equation} Thus, T=0 \begin{equation} ⟨TT^*v,v⟩=⟨T^*v,T^*v⟩ \end{equation} \begin{equation} T=(T^*)^*=0 \end{equation} Then, we see that this also implies T=0. \end{document}

Problem 6.3.11

Joshua Hoover

September 2025

0 = < T^{*} 𝑇𝑣, v > = < 𝑇𝑣, 𝑇𝑣 >

(1)

Thus, T=0

< T T^{*} v, v > = < T^{*} v, T^{*} v >

(2)

T = {(T^{*})}^{*} = 0

(3)

Then, we see that this also implies T=0.