Since a bounded linear operator T from the normed space X to normed space Y is defined and given as; \forall x\in D\left( T\right) \exists k>0, such that \left\Vert Tx\right\Vert \leq k\left\Vert x\right\Vert , then for any \alpha \in F, % \left\Vert \alpha T\right\Vert =

• A) \underset{\underset{\left\Vert x\right\Vert =1}{x\in D(T)}}{\sup }\alpha \left\Vert Tx\right\Vert
• B) \underset{\underset{x\neq 0}{x\in D(T)}}{\sup }\frac{\alpha \left\Vert Tx\right\Vert }{\left\Vert x\right\Vert }
• C) \underset{\underset{\left\Vert X\right\Vert =1}{x\in D(T)}}{\sup }\left( -\left\vert \alpha \right\vert \right) \left\Vert Tx\right\Vert
• D) \underset{\underset{\left\Vert x\right\Vert =1}{x\in D(T)}}{\sup }% \left\vert \alpha \right\vert \left\Vert Tx\right\Vert

On a normed space $X$ of all polynomials of form $x\left( t\right) =t^{n+1},n\in %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion $ defined on $\left[ -1,1\right] ,$if $D$ is a differential linear operator defined and given as; $D\left( x\left( t\right) \right) =\left\{ \frac{d}{dt}x\left( t\right) :x\left( t\right) \in P\left[ -1,1\right] ,-1\leq t\leq 1\right\} ,$ then $% \left\Vert D\right\Vert =$__________.

• A) $1$
• B) $n+1$
• C) $0$
• D) $n$

Let $T:X\rightarrow Y$ be an invertible linear operator from $X$ to $ Y$, where $\dim X,\dim Y<\infty ,$ then

• A) $\dim X>\dim Y$
• B) $\dim X<\dim Y$
• C) $\dim X=\dim Y$
• D) $\dim X=-\dim Y$

Which of the following is/are bounded linear operator

• A) Identity operator .
• B) Zero operator.
• C) Integral operator.
• D) All given options.

If$~T_{1}$ and $T_{2}$ are linear operators from $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ into $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $, defined by $T_{1}\left( x\right) =-x$ and $T_{2}\left( x\right) =x,\forall x\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ,$ then $\left( T_{1}T_{2}\right) ^{-1}\left( x\right) =$_________.

• A) $\frac{1}{x}$
• B) $-\frac{1}{x}$
• C) $-x$
• D) $x$

Operator $T:\,D(T) \to Y$ is bounded if there is a real number c such that for all $x \in \,D(T)\,,$

• A) $\left\| {Tx} \right\| \leqslant \,c\left\| x \right\|\,$
• B) $\left\| {Tx} \right\| \geqslant \,c\left\| x \right\|$
• C) $\left\| {Tx} \right\| \leqslant \,\left\| x \right\|$
• D) $\left\| {Tx} \right\| = \,c\left\| x \right\|$

If $T:% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2}\rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2}$ defined by $T(x,y)=\left( y,-5x+4y\right) ,$ then $T^{-1}\left( -2,7\right) =$_______

• A) (-3,-2)
• B) (3,2)
• C) (-3,2)
• D) (3,-2)

If $Tx_{n}=0$ for a sequence $\left\{ x_{n}\right\} $ in a null space $% N\left( T\right) ~$of a bounded linear operator $T$ on a normed space $X,$ then$~x_{n}\rightarrow x\Longrightarrow x=$

• A) unit vector
• B) is neither unit vector nor zero vector
• C) zero vector
• D) is either unit vector or zero vector

If$~T_{1}$ and $T_{2}$ are bounded linear operators from the normed space $X$ into $Y$, then $\left\Vert T_{1}+T_{2}\right\Vert $________.

• A) All options are equivalent
• B) $=\underset{\underset{\left\Vert x\right\Vert =1}{x\in D(T)}}{\sup }% \left\Vert \left( T_{1}+T_{2}\right) x\right\Vert $
• C) $=\underset{\underset{\left\Vert x\right\Vert =1}{x\in D(T)}}{\sup }% \left\Vert T_{1}x+T_{2}x\right\Vert $
• D) $\leq \underset{\underset{\left\Vert x\right\Vert =1}{x\in D(T)}}{\sup }% \left\Vert T_{1}x\right\Vert +\underset{\underset{\left\Vert x\right\Vert =1}% {x\in D(T)}}{\sup }\left\Vert T_{2}x\right\Vert $

If $\left\{ x_{n}\right\} $ is a sequence in a null space $N\left( T\right) ~$of a bounded linear operator $T$ on a normed space $X,$ then

• A) $Tx_{n}=~$unit vector
• B) $Tx_{n}$ is neither unit vector nor zero vector
• C) $Tx_{n}$ is either unit vector or zero vector
• D) $Tx_{n}=$ zero vector

Let $T:\left[ 0,\infty\right) \rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ be defined by $T\left( x\right) =x$, then its extension $\widetilde{T}$ on $M=% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ is

• A) $\widetilde{T}(x)=\frac{x+\left\vert x\right\vert }{2}$
• B) $\widetilde{T}(x)=x$
• C) All above are valid
• D) $\widetilde{T}(x)=\left\vert x\right\vert $

Two operators {T_1}\,\,and\,\,{T_2} are equal if

• A) D({T_1}) = D({T_2}) and \forall \,\,x \in D({T_1}) = D({T_2})\,\, \Rightarrow \,\,{T_1}(x) = {T_2}(x)
• B) \forall \,\,x \in D({T_1}) = D({T_2})\,\, \Rightarrow \,\,{T_1}(x) = {T_2}(x)
• C) D({T_1}) = D({T_2})
• D) D({T_1}) = D({T_2}) and \forall \,\,x \in R({T_1}) = R({T_2})\,\, \Rightarrow \,\,{T_1}(x) = {T_2}(x)

The mapping; $T:V\rightarrow V$ defined by $T(v)=a+v$, where $\ v\in V~$ is linear if

• A) $a=0$
• B) $a\neq 0$
• C)
• D)

Since a bounded linear operator $T$ from the normed space $X$ to normed space $Y$ is defined and given as; $\forall x\in D\left( T\right) $ $\exists $ $k>0,$ such that $\left\Vert Tx\right\Vert \leq k\left\Vert x\right\Vert ,$ then the minimum value of $k$ is ___________.

• A) $\underset{\underset{X\neq 0}{x\in D(T)}}{\sup }\frac{\left\Vert Tx\right\Vert }{\left\Vert x\right\Vert }$
• B) any arbitrary non negative real number
• C) not defined
• D) $\underset{\underset{X\neq 0}{x\in D(T)}}{\inf }\frac{\left\Vert Tx\right\Vert }{\left\Vert x\right\Vert }$

If $T:% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2}\rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{3}~$is defined by $T\left( x,y\right) =\left( y-x,y,x-1\right) ,$ then the value of $T\left( 0,0\right) \Longrightarrow $

• A) $T^{-1}$ exists
• B) $T~$ is not linear
• C) $T^{-1}$ does not exists
• D) $T$ is linear

If T:% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2}\rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{3}~is defined by T\left( x,y\right) =\left( y-x,y,x-1\right) , then the value of T\left( 0,0\right) \Longrightarrow

• A) T^{-1} does not exists
• B) T is linear
• C) T~ is not linear
• D) T^{-1} exists

If Tx_{n}=0 for a sequence \left\{ x_{n}\right\} in a null space % N\left( T\right) ~of a bounded linear operator T on a normed space X, then~x_{n}\rightarrow x\Longrightarrow x=

• A) is either unit vector or zero vector
• B) zero vector
• C) unit vector
• D) is neither unit vector nor zero vector

Let T:D\left( T\right) \rightarrow Y be a bounded linear operator from normed space X to Banach space Y, and \forall x\in \overline{D\left( T\right) }~\exists ~a sequence \left\{ x_{n}\right\} in D(T) such that x_{n}\rightarrow x,then~\left\Vert Tx_{m}-Tx_{n}\right\Vert \Longrightarrow

• A) \left\{ Tx_{n}\right\} ~is a Cauchy Sequence in Y
• B) \left\{ Tx_{n}\right\} ~is a Cauchy Sequence in \overline{Y}
• C) \left\{ Tx_{n}\right\} ~is a Cauchy Sequence in \overline{R(T)}
• D) \left\{ Tx_{n}\right\} ~is not a Cauchy Sequence in Y

For a fixed k=\left( k_{1},k_{2}\right) , defining the linear functional % f:% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2}\rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion as f\left( x\right) =x.k=x_{1}k_{1}+x_{2}k_{2},~\forall \left( x_{1},x_{2}\right) \in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2}, then \left\Vert f\right\Vert =

• A) \min \left( x_{1}k_{1},x_{2}k_{2}\right)
• B) \left\Vert k\right\Vert
• C) \left\Vert x\right\Vert
• D) \max \left( x_{1}k_{1},x_{2}k_{2}\right)

If \left\{ x_{n}\right\} is a sequence in a null space N\left( T\right) ~of a bounded linear operator T on a normed space X, then

• A) Tx_{n} is neither unit vector nor zero vector
• B) Tx_{n} is either unit vector or zero vector
• C) Tx_{n}= zero vector
• D) Tx_{n}=~unit vector