Dual space of l^1 is

• A) {\ R}
• B) {\ R}^n
• C) l^1
• D) l^\infty \,

Dual space of {\ R}^n is

• A) {\Bbb C}^n
• B) {\ R}^n
• C) {\Bbb Z}^n
• D) {\ R}

For an inner product space defined on a real vector space \left\langle {x,y} \right\rangle = .........

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

This following expression {\mkern 1mu} {\left\| {x + y} \right\|^2} + {\left\| {x - y} \right\|^2} = 2({\left\| x \right\|^2} + {\left\| y \right\|^2}) is called........

• A) Parallelogram equality
• B) Pythagorean theorem
• C) Cauchy schwarz inequality
• D) Apollonius identity

The pair (V, < .\,,\,. > ) is called

• A) Inner product space.
• B) Banach space.
• C) Metric space.
• D) Complete space.

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) \Longrightarrow \left\langle x_{n},y_{n}\right\rangle =2\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) \nRightarrow \left\langle x_{n},y_{n}\right\rangle \longrightarrow< \left\langle x,y\right\rangle

<p>In an Inner Product space say X,~if the sequences \left\{ x_{n}\right\} and \left\{ y_{n}\right\} are Cauchy, then \left\langle<x_{n},y_{n}\right\rangle is ---------.</p>

• A) none of these
• B) may or may not a Cauchy Sequence
• C) not necessarily a Cauchy Sequence
• D) necessarily a Cauchy Sequence

Let \left( {V,\left\langle {.,.} \right\rangle } \right) be an inner product space over a field F, then ......

• A) \left\langle {x,\alpha .y} \right\rangle = \bar \alpha \left\langle {y,x} \right\rangle ,\,\,\,\,\,\,\,\,\,\,\,\forall x,y \in V,\alpha \in F.
• B) \left\langle {x,\alpha .y} \right\rangle = \alpha \left\langle {x,y} \right\rangle ,\,\,\,\,\,\,\,\,\,\,\,\forall x,y \in V,\alpha \in F.
• C) \left\langle {x,\alpha .y} \right\rangle = \bar \alpha \left\langle {x,x} \right\rangle ,\,\,\,\,\,\,\,\,\,\,\,\forall x,y \in V,\alpha \in F.
• D) \left\langle {x,\alpha .y} \right\rangle = \bar \alpha \left\langle {x,y} \right\rangle ,\,\,\,\,\,\,\,\,\,\,\,\forall x,y \in V,\alpha \in F.

For an inner product space {\mkern 1mu} {\left\| {x + y} \right\|^2} + {\left\| {x - y} \right\|^2} = ...........

• A) 2({\left\| x \right\|^2} + {\left\| y \right\|^2})
• B) {\left\| x \right\|^2}{\left\| y \right\|^2}
• C) {\left\| x \right\|^2} - {\left\| y \right\|^2}
• D) {\left\| x \right\|^2} + {\left\| y \right\|^2}

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 >

<p>In an Inner Product space say $X,~$if the sequences $\left\{ x_{n}\right\} $ and $\left\{ y_{n}\right\} $ are Cauchy, then $\left\langle<x_{n},y_{n}\right\rangle $ is ---------.</p>

• A) may or may not a Cauchy Sequence
• B) not necessarily a Cauchy Sequence
• C) necessarily a Cauchy Sequence
• D) none of these

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

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

||z - x|{|^2} + ||z - y|{|^2} = \frac{1}{2}||x - y|{|^2} + 2||z - \frac{1}{2}(x + y)|{|^2} is called………..

• A) Parallelogram equality
• B) Polarization identity
• C) Apollonius identity
• D) Pythagorean theorem

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) \[\alpha \left\langle {x,y} \right\rangle \]
• C) \[\alpha \left\langle {y,x} \right\rangle \]
• D) \[\alpha \left\langle { - x,y} \right\rangle \]

Dual space of $$ l^1 $$ is

• A) $$ {\ R}^n $$
• B) $$ l^\infty \, $$
• C) $$ {\ R} $$
• D) $$ l^1 $$

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

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

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

The pair $$ (V, < .\,,\,. > ) $$ is called

• A) Inner product space.
• B) Metric space.
• C) Banach space.
• D) Complete space.

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

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