The dual space of a normed is ........ norm space.

• A) an incomplete
• B) a complete
• C)
• D)

Which of the following is not a condition of an inner product space?

• A) $\left\langle {\alpha x + y,z} \right\rangle = \left\langle {x,z} \right\rangle + \alpha \left\langle {y,z} \right\rangle$
• B) $\left\langle {x,y} \right\rangle = \overline {\left\langle {y,x} \right\rangle }$
• C) $$\left\langle {x,x} \right\rangle \geqslant 0$$
• D) $$\left\langle {\alpha x,y} \right\rangle = \alpha \left\langle {x,y} \right\rangle$$

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

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

An element x of an inner product space X is said to be orthogonal to an element $$y \in X$$ if.......

• A) $\left\langle {x,\left. y \right\rangle } \right. > 0$
• B) none of these
• C) $\left\langle {x,\left. y \right\rangle } \right. = 0$
• D) $\left\langle {x,\left. y \right\rangle } \right. < 0$

Norm in $$V\,\,{\text{for}}\,\,x\, \in V$$ is defined as

• A) $$\left\| x \right\| = \sqrt { < x\,,\,x > } \,$$
• B) $$\left\| x \right\| = \left| { < x\,,\,x > } \right|^2$$
• C) $$\left\| x \right\| = \sqrt { < x/2\,,\,x/2 > }$$
• D) $$\left\| x \right\| = < x\,,\,x >$$

In an Inner Product space say $X,~$for any sequences $\left\{ x_{n}\right\}$ and $\left\{ y_{n}\right\} ,$ if $x_{n}\longrightarrow x$ and $y_{n}\longrightarrow y$, then --------.

• A) $\nRightarrow \left\langle x_{n},y_{n}\right\rangle \longrightarrow< \left\langle x,y\right\rangle$
• B) $\Longrightarrow \left\langle x_{n},y_{n}\right\rangle \longrightarrow \left\langle x,y\right\rangle$
• C) $\Longrightarrow \left\langle x_{n},y_{n}\right\rangle \neq \left\langle x,y\right\rangle$
• D) $\Longrightarrow \left\langle x_{n},y_{n}\right\rangle =2\left\langle x,y\right\rangle$

An Isomorphism of a normed space X to a normed space Y is a ................operator that preserves norm.

• A) non linear
• B) bijective linear
• C) surjective linear
• D) injective linear

In an inner product space $X$ over the field $F$,$\langle x ,z \rangle = \langle y ,z \rangle$

• A) $\Rightarrow$x$\neq$y, for all z$\in X$
• B) $\Rightarrow$x=y, for some z$\in X$
• C) ⇒x≠y, for all z∈X
• D) $\nRightarrow$x=y, for all z$\in X$

An element x of an inner product space X is said to be ……to an element $$y \in X$$ if $\left\langle {x,\left. y \right\rangle } \right. = 0$

• A) orthonormal
• B) orthogonal
• C) parallel
• D) equal

Every Inner Product space is a Metric Space as well.

• A) True
• B) False
• C)
• D)

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

Every Inner product space is a metric space with norm given by;

• A) $d(x,y)=\sqrt{\left\langle x-y,x-y\right\rangle }$
• B) $\left\Vert x-y\right\Vert =\sqrt{\left\langle x-y,x-y\right\rangle }$
• C) $d(x,y)=\left\Vert x-y\right\Vert$
• D) All above are equivalent

A hilbert space is a /an...........

• A) complete Inner product space
• B) Incomplete norm space
• C) Incomplete Inner product space
• D) complete norm space

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

Operator T:\,D(T) \to Y is bounded if there is a real number c such that for all x \in \,D(T)\,,

• A) \left\| {Tx} \right\| = \,c\left\| x \right\|
• B) \left\| {Tx} \right\| \geqslant \,c\left\| x \right\|
• C) \left\| {Tx} \right\| \leqslant \,c\left\| x \right\|\,
• D) \left\| {Tx} \right\| \leqslant \,\left\| x \right\|

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~T_{1} and T_{2} are bounded linear operators from the normed space X into Y, then \left\Vert T_{1}+T_{2}\right\Vert________.

• A) None of these
• B) =\underset{\underset{\left\Vert x\right\Vert =1}{x\in D(T)}}{\sup }% \left\Vert T_{1}x\right\Vert +\underset{\underset{\left\Vert X\right\Vert =1}% {x\in D(T)}}{\sup }\left\Vert T_{2}x\right\Vert
• C) \leq \underset{\underset{\left\Vert x\right\Vert =1}{x\in D(T)}}{\sup }% \left\Vert T_{1}x+T_{2}x\right\Vert
• D) \geq \underset{\underset{\left\Vert x\right\Vert =1}{x\in D(T)}}{\sup }% \left\Vert \left( T_{1}+T_{2}\right) 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 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) bounded
• B) not necessarily
• C) essentially
• D) never

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