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

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\} $ converges in $\overline{R(T)}$
• B) $\left\{ Tx_{n}\right\} $ converges in $Y$
• C) $\left\{ Tx_{n}\right\} $ converges in $\overline{Y}$
• D) $\left\{ Tx_{n}\right\} ~$may not converges in $Y$

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

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)

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

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) $T_{1}-T_{2}$ is a zero operator
• B) $T_{1}-T_{2}$ is not necessarily a zero operator
• C)
• D)

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

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

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 f\right\vert \left\vert x\right\vert $
• C) $\left\Vert x\right\Vert \left\Vert k\right\Vert ~$
• D) $\left\Vert f\right\Vert \left\Vert k\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 $\ \left\Vert T\right\Vert =$__________.

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

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) $k^{2}$
• C) $0$
• D) $\frac{1}{k}$

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

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

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

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

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 not one-one
• B) $T$ is one-one
• 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 maximum 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) $\underset{\underset{X\neq 0}{x\in D(T)}}{\sup }\frac{\left\Vert Tx\right\Vert }{\left\Vert x\right\Vert }$

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 $x\in X$ $\exists ~\left\{ \alpha _{i}\right\} _{i=1}^{n},$ $x=\sum_{i=1}^{n}% \alpha _{i}e_{i},$ then $\left\Vert T\left( \sum_{i=1}^{n}\alpha _{i}e_{i}\right) \right\Vert $

• A) $\leq \sum_{i=1}^{n}\left\vert T\alpha _{i}\right\vert \left\Vert e_{i}\right\Vert $
• B) $\leq \sum_{i=1}^{n}\left\vert T\alpha _{i}\right\vert \left\Vert Te_{i}\right\Vert $
• C) $\leq T\left\Vert \sum_{i=1}^{n}\alpha _{i}e_{i}\right\Vert $
• D) $\leq \sum_{i=1}^{n}\left\vert \alpha _{i}\right\vert \left\Vert Te_{i}\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) $1$
• C) $0$
• D) $\underset{t\in \left[ 0,1\right] }{\max }x\left( t\right) $

For an element x belongs to an inner product space , \left\langle {x,x} \right\rangle = ......... if and only if x=0.

• A) 0
• B) infinity
• C) less than 0
• D) greater than 0

The following expression represents ……… inequality \frac{{|x - z|}}{{1 + |x - z|}} \leqslant \frac{{|x - y|}}{{1 + |x - y|}} + \frac{{|y - z|}}{{1 + |y - z|}}

• A) Rectangle inequality
• B) Triangular inequality
• C) Minkowski’s inequality
• D) Holder’s inequality