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 $

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

• A) \left\Vert x\right\Vert \left\Vert k\right\Vert ~
• B) \left\vert x\right\vert \left\vert k\right\vert
• C) \left\vert f\right\vert \left\vert x\right\vert
• D) \left\Vert f\right\Vert \left\Vert k\right\Vert

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) -x
• C) \frac{1}{x}
• D) x

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 maximum value of k is__________.

• A) \underset{\underset{X\neq 0}{x\in D(T)}}{\inf }\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)}}{\sup }\frac{\left\Vert Tx\right\Vert }{\left\Vert x\right\Vert }

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

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 =0\Longrightarrow $

• A) $x\in \{0\}$
• B) $k=0$
• C) $k<0$
• D) $T$ is a zero operator

If the integral operator $I:c\left[ 0,1\right] \rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ,$ on the space of all contnuous functions on $\left[ 0,1\right] $ defined by $f(x)=\int_{0}^{1}x\left( t\right) dt$, is a linear functional, then $% \left\vert f\left( x\right) \right\vert \leq $

• A) $\left\vert x\right\vert $
• B) $1$
• C) $\left\Vert x\right\Vert $
• D) $\underset{0\leq t\leq 1}{\max }\left\vert x\left( t\right) \right\vert $

If T:c[0,1]→c[0,1], which is defined and given as; T(x)={∫₀¹κ(t,τ)x(τ)dτ:|κ|<k₀},then ∀ x ∈ c[0,1] ∃ k>0 such that ‖Tx‖≤k‖x‖⇒

• A) T is linear bounded
• B) T is non-linear bounded
• C) T is non-linear unbounded
• D) T is linear unbounded

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.If D\left( T\right) =\{0\}, then \left\Vert T\right\Vert =

• A) \infty
• B) 0
• C) \frac{1}{k}
• D) k^{2}

Let T:X \to Y be a linear operator, then restriction of T is expressed as

• A) {T_{\left| B \right.}}:B \to B\,\,\,\,,\,B \subseteq X
• B) {T_{\left| B \right.}}:B \to Y\,\,\,,\,\,B \subseteq X
• C) {T_{\left| B \right.}}:X \to B\,\,\,,\,\,B \subseteq Y
• D) {T_{\left| B \right.}}:Y \to B\,\,,\,\,B \subseteq Y

On a normed space $X$ of all $n$ degree polynomials defined on $\left[ -1,1% \right] ,$if $D$ is a differential linear operator defined and given as; $D\left( x\left( t\right) \right) =\left\{ \frac{d}{dt}x\left( t\right) :x\left( t\right) \in X,-1\leq t\leq 1\right\} ,$ then $\forall x\in X$ $% \exists $ $k>0$ such that $\left\Vert Dx\right\Vert \leq k\left\Vert x\right\Vert \Longrightarrow $

• A) $D$ is non-linear unbounded
• B) $D$ is linear unbounded
• C) $D$ is a non-linear bounded
• D) $D$ is a linear bounded

On a normed space X, for the identity operator I:X\rightarrow X,\left\Vert I\right\Vert =\underset{\underset{x\neq 0}{x\in D\left( I\right) }}{\sup }% \frac{\left\Vert Ix\right\Vert }{\left\Vert x\right\Vert }=__________.

• A) 1
• B) \left\Vert x\right\Vert
• C) \left\Vert kx\right\Vert ,k>0
• D) 0

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)

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

If a sequence x_{n}\rightarrow x in a normed space X, then for a bounded linear operator T on X, then

• A) Tx_{n}\nrightarrow Tx
• B) Tx_{n}\rightarrow Tx
• C)
• D)

On a normed space X, for the zero operator $O:X\rightarrow X,\left\Vert O\right\Vert =\underset{\underset{x\neq 0}{x\in D\left( I\right) }}{\sup }% \frac{\left\Vert Ox\right\Vert }{\left\Vert x\right\Vert }=$________

• A) $\left\Vert kx\right\Vert ,k>0$
• B) 1
• C) $\left\Vert x\right\Vert $
• D) 0

On a normed space X of all n degree polynomials 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 X,-1\leq t\leq 1\right\} , then \forall x\in X % \exists k>0 such that \left\Vert Dx\right\Vert \leq k\left\Vert x\right\Vert \Longrightarrow

• A) D is a linear bounded
• B) D is non-linear unbounded
• C) D is linear unbounded
• D) D is a non-linear bounded

If A:% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2}\rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2} is defined as Ax=y~and given by; \left( \begin{array}{cc} \alpha _{11} & \alpha _{12} \\ \alpha _{21} & \alpha _{22}% \end{array}% \right) \left( \begin{array}{c} \xi _{1} \\ \xi _{2}% \end{array}% \right) =\left( \begin{array}{c} \eta _{1} \\ \eta _{2}% \end{array}% \right) \Longrightarrow \eta _{j}=\sum_{i=1}^{2}\alpha _{ji}\xi _{i}, then % A is

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

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) n+1
• B) 0
• C) n
• D) 1