Every complete Inner product space is ---------.

• A) Banach space
• B) Hilbert space
• C) Euclidean space
• D) Complex space

which of the following space is not an inner product space?

• A) $Euclidean\,\,space\,\,{R^n}$
• B) $Unitary\,\,space\,\,{C^n}$
• C) the space ${l^{p\,}}\,with\,p \ne 2$
• D) the space ${l^2}$

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) ${\left\| x \right\|^2} + {\left\| y \right\|^2}$

Let $\left( {V,\left\langle {.,.} \right\rangle } \right)$ be an inner product space over a field F, then.............$${\text{\& }}\forall x,y,z \in V,\alpha ,\beta \in F$$

• A) $$\left\langle {\alpha x + \beta y,z} \right\rangle = \alpha \left\langle {x,z} \right\rangle + \beta \left\langle {y,z} \right\rangle$$
• B) $$\left\langle {\alpha x + \beta y,z} \right\rangle = \alpha \left\langle {x,z} \right\rangle /\beta \left\langle {y,z} \right\rangle$$
• C) $$\left\langle {\alpha x + \beta y,z} \right\rangle = \alpha \left\langle {x,z} \right\rangle - \beta \left\langle {y,z} \right\rangle$$
• D) $$\left\langle {\alpha x + \beta y,z} \right\rangle = \alpha \left\langle {x,z} \right\rangle *\beta \left\langle {y,z} \right\rangle$$

For an inner product space <x+y,z>=………..

• A) None of these
• B) <x,z> . <y,z>
• C) <x,z> - <y,z>
• D) <x,z> + <y,z>

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

In an inner product space $X$ over the field of Real numbers,$\text{for all x,y and z \in X\ and }\alpha \in F$ $\text{,then }\text{} \langle \text{}\alpha x +\beta y ,z \rangle =$

• A) $\alpha \langle x ,z \rangle +\overline{\beta } \langle y ,z \rangle$
• B) $\alpha \langle x ,z \rangle +\beta \langle y ,z \rangle$
• C)
• D)

In an inner product space $X$ over the field of Complex numbers, $\text{for all x,y \in X\ and }\alpha \in F\text{, then }\text{} \langle x ,\alpha y \rangle =\text{}$

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

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

…………… is algebraically reflexive.

• A) A dual space of a vector space
• B) An infinite dimensional vector space
• C) A finite dimensional vector space
• D) Canonical mapping

Dual space of a normed space is

• A) Banach space.
• B) functional.
• C) Metric space.
• D) Incomplete.

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

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

Null space is also a

• A) Metric space.
• B) Vector space.
• C) Linear functional.
• D) Canonical mapping.

<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

Dual space of $${\ R}^n$$ is

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

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

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

Zero vector lemma applies on

• A) Infinite dimensional vector space.
• B) Dual space of a vector space.
• C) Finite dimensional vector space.
• D) Canonical mapping.

Dual space of $$l^1$$ is

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

Which of the following is 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) $$\left\langle {\alpha x,y} \right\rangle = \left\langle {x,\alpha y} \right\rangle$$
• C) $\left\langle {x,y} \right\rangle = \overline {\left\langle {x,y} \right\rangle }$
• D) $$\left\langle {x,x} \right\rangle \geqslant 0$$

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

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