T is a linear operator . If {T^{ - 1}} exists, it is a

• A) discontinuous operator.
• B) non linear operator.
• C) linear operator.
• D) Zero operator.

If T is a bounded linear operator on a normed space X, then for the Null space N(T);

• A) N(T)\nsubseteq \overline{N(T)}
• B) N(T)\subseteq \overline{N(T)}
• 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 \ \left\Vert T\right\Vert =__________.

• A) both \underset{\underset{\times \neq 0}{x\in D(T)}}{\sup }\frac{\left\Vert Tx\right\Vert }{\left\Vert x\right\Vert } and \underset{\underset{% \left\Vert x\right\Vert =1}{x\in D(T)}}{\sup }\left\Vert Tx\right\Vert
• B) \underset{\underset{\times \neq 0}{x\in D(T)}}{\sup }\frac{\left\Vert Tx\right\Vert }{\left\Vert x\right\Vert }
• C) None of these
• D) \underset{\underset{\left\Vert x\right\Vert =1}{x\in D(T)}}{\sup }% \left\Vert Tx\right\Vert

Let $B\left( {X,Y} \right)$ be the set of all ……………operators from a normed space X to a normed space Y. If Y is a Banach space, then $B\left( {X,Y} \right)$ is a Banach space.

• A) Non linear
• B) Bounded linear
• C) Unbounded linear
• D) Linear

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\left( x\right) \right\vert =\left\vert x.k\right\vert \leq

• A) \underset{x\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2}}{\max }\left\vert f\left( x\right) \right\vert
• B) \left\Vert x\right\Vert \left\Vert k\right\Vert
• C) \underset{x\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion }{\max }\left\Vert x\right\Vert
• D) \left\vert x\right\vert \left\vert k\right\vert

If T_{1} and T_{2} are equal operators defined on a normed space X, then for any x\in X,T_{1}x=T_{2}x\Longrightarrow

• A) x\neq 0 necessarily
• B) x=0.
• C)
• D)

Let X be a normed space, f :X \rightarrow \mathbb{R} be the linear functional and for any \alpha \in \mathbb{R} , then \alpha f :X \rightarrow \mathbb{R} defined by \left (\alpha f\right )(x) =\alpha f(x) \forall x\text{} \in X\text{, is _______} a linear functional.

• A) essentially
• B) bounded
• C) never
• D) not necessarily

If T:% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2}\rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ~is defined by T\left( x,y\right) =y-x, then T^{-1}\left( 6\right) =\left\{ \left( t,t-6\right) :t\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \right\} \Longrightarrow

• A) T is one-one
• B) T is not one-one
• C)
• D)

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)

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 T_{2}^{-1}T_{1}^{-1}\left( x\right) =_______.

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

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) All above are valid
• C) \widetilde{T}(x)=x
• D) \widetilde{T}(x)=\left\vert x\right\vert

For a fixed t\in \left[ 0,1\right] , defining the linear functional on the class of all continous functions on \left[ 0,1\right] , f:c\left[ 0,1% \right] \rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion as f\left( x\right) =x\left( t\right) ,~\forall x\in c\left[ 0,1\right] , then \left\Vert f\right\Vert =

• A) \underset{t\in \left[ 0,1\right] }{\min }x\left( t\right)
• B) 0
• C) 1
• D) \underset{t\in \left[ 0,1\right] }{\max }x\left( t\right)

If T is a linear opertor on a finite dimensional normed space % X having basis \left\{ e_{1},e_{2},\ldots ,e_{n}\right\} ,then for any % x\in X \exists ~\left\{ \alpha _{i}\right\} _{i=1}^{n}\subset F, such that

• A) x=\sqrt{\sum_{i=1}^{n}\alpha _{i}e_{i}}
• B) x=\sum_{i=1}^{n}\alpha _{i}e_{i}
• C) x=\alpha _{i}\sum_{i=1}^{n}e_{i}, for any fixed \alpha _{i}
• D) x=e_{i}\sum_{i=1}^{n}\alpha _{i}, for any fixed e_{i}

Let T:D\left( T\right) \rightarrow Y be a linear operator from normed space X to normed space Y, then

• A) \forall x\in \overline{D\left( T\right) }~\exists ~a sequence \left\{ x_{n}\right\} in D(T) such that x_{n}\rightarrow x
• B) \nexists x\in \overline{D\left( T\right) }~\forall sequences \left\{ x_{n}\right\} in D(T) such that x_{n}\rightarrow x
• C) \forall x\in \overline{D\left( T\right) }~\nexists ~a sequence \left\{ x_{n}\right\} in D(T) such that x_{n}\rightarrow x
• D) \exists x\in \overline{D\left( T\right) }~\forall sequences \left\{ x_{n}\right\} in D(T) such that x_{n}\rightarrow x

If a normed space X is finite dimensional , then every linear operator on X is

• A) Continuous.
• B) Differentiable.
• C) Bounded.
• D) Integrable.

Let S and T are linear operators , we have

• A) {(ST)^{ - 1}} = {T^{ - 1}}{S^{ - 1}}
• B) {(ST)^{ - 1}} = ST
• C) {(ST)^{ - 1}} = {S^{ - 1}}{T^{ - 1}}
• D) {(ST)^{ - 1}} = I

Let B\left( {X,Y} \right) be the set of all bounded linear operators from a normed space X to a normed space Y. If Y is a Banach space, then B\left( {X,Y} \right) is a………..

• A) None of these
• B) Norm
• C) Hilbert
• D) Banach

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

Let T:% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion be defined by T\left( x\right) =x^{2}, then the restriction T_{|A} is one-one on A=

• A) \left( -\infty ,\infty \right)
• B) \left( -1,\infty \right)
• C) \left( -\infty ,1\right)
• D) \left( -\infty ,0\right)

For a bounded linear operator T , the null space {\rm N}(T) is

• A) unbounded.
• B) bounded.
• C) closed.
• D) open.