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

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

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

A linear functional f with domain D in a normed space is continuous if and only if f is

• A) Integrable.
• B) Bounded.
• C) Differentiable.
• D) Real valued.

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

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

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

............ is a mapping from a space 'X' to its Second Dual.

• A) Isomorphism
• B) Surjection
• C) Canonical
• D) Bijection

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

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

Functionals defined on C[a,b] are

• A) Integrable.
• B) Canonical mappings.
• C) Linear and bounded.
• D) Unbounded.

Canonical mapping is defined as

• A) $$C:X \to X^{**}$$
• B) $$C:X \to R$$
• C) $$C:X \to X$$
• D) $$C:X \to R(Y)$$

Matrix operator is

• A) Linear and bounded.
• B) Non linear.
• C) Linear.
• D) Nonlinear and unbounded.

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

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

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

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

• A) $\leq \sum_{i=1}^{n}\left\vert \alpha _{i}\right\vert \left\Vert Te_{i}\right\Vert$
• B) $\leq \sum_{i=1}^{n}\left\vert T\alpha _{i}\right\vert \left\Vert e_{i}\right\Vert$
• C) $\leq T\left\Vert \sum_{i=1}^{n}\alpha _{i}e_{i}\right\Vert$
• D) $\leq \sum_{i=1}^{n}\left\vert T\alpha _{i}\right\vert \left\Vert Te_{i}\right\Vert$

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 $% x\in X$ $\exists ~\left\{ \alpha _{i}\right\} _{i=1}^{n}\subset F,$ such that

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

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) zero vector
• C) is neither unit vector nor zero vector
• D) is either unit vector or 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) $\widetilde{T}(x)=x$
• B) $\widetilde{T}(x)=\frac{x+\left\vert x\right\vert }{2}$
• C) All above are valid
• D) $\widetilde{T}(x)=\left\vert x\right\vert$

If a sequence xn→x in a normed space X, then for a bounded linear operator T on X_______.

• A) limn→∞‖Txn-Tx‖>0
• B) ‖Txn-Tx‖≠0
• C) limn→∞‖Txn-Tx‖<0
• D) ‖Txn-Tx‖≤0

Dot product is a

• A) operator.
• B) Mapping.
• C) Function.
• D) Functional.

For all x, y belongs to an an inner product space$$\left\langle {\alpha x,y} \right\rangle = .............\,$$

• A) $$\alpha \left\langle { - x,y} \right\rangle$$
• B) $$\alpha \left\langle {y,x} \right\rangle$$
• C) $$\alpha \left\langle {x, - y} \right\rangle$$
• D) $$\alpha \left\langle {x,y} \right\rangle$$