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 $Y$

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

• A) $N(T)\subseteq \overline{N(T)}$
• B) $N(T)\nsubseteq \overline{N(T)}$
• C)
• D)

A canonical mapp is said to be ......... if it is surjective.

• A) transitive
• B) isometry
• C) algebraically reflexive
• D) symmetric

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)

Which of the following an example of Linear Functional?

• A) 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 $I\left( f\right) =\int_{0}^{1}f\left( t\right) dt$
• B) the derivative operator on space of all real polynomials
• C)
• D)

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

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

Linear Functional f is a linear operator from a normed space X to ________.

• A) a Banach space Y i.e. f:X→Y
• B) range of f i.e f:X→R(f)⊆Y
• C) any normed space Y i.e. f:X→Y
• D) scalar field F of given vector space X(F) i.e. f:X→F

For n dimensional vector space and its dual space we have

• A) $$\dim (X) \subset \,\dim (X^ * )$$
• B) $$R(X) = X^ * \, = n\,$$
• C) $$D(X) = X^ * = n$$
• D) $$\dim (X) = \dim (X^ * ) = n\,$$

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

Let X and Y be normed spaces. A linear operator T:X→Y is said to be bounded if ------------ such that ‖Tx‖≤k‖x‖.

• A) ∄ k>0,∀ x∈X
• B) ∃ k>0, ∀ x∈X
• C) ∀ k>0,∃ x∈ X
• D) ∀ k>0, ∄ x∈X

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

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

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}$ and $g :X \rightarrow \mathbb{R}$ be the linear functionals, then $f +g :X \rightarrow \mathbb{R}$ defined by $\left (f +g\right )(x) =f(x) +g(x) \forall x \in X ,\text{ is_______}$ a linear functional.

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

Which of the following is an example of Linear Functional?

• A) the determinant function $D:V\rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion$ from the space of n-square matrices to $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion$
• B) the projection function $\pi _{i}:% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n}\rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion$ defined by $\pi _{i}\left( x_{1},x_{2},\ldots ,x_{i},\ldots x_{n}\right) =x_{i}$
• C)
• D)

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

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

For algebraically reflexive mapping

• A) $$R(C) = X^{ * * } \,\,$$
• B) $$X = X^{ * * }$$
• C) $$D(C) = X^{ * * }$$
• D) $$X \subset X^{ * * }$$

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)

If T is a bounded linear operator from a normed space X to normed space Y, then T is _________.

• A) continuous as well irrespective of dimension of X and Y
• B) continuous as well provided that X and Y are finite dimensional
• C) unbounded as well provided that X and Y are infinite dimensional
• D) unbounded as well irrespective of dimension of X and Y