1.3 Inner product spaces and hilbert spaces

 Page 1 / 1
Review of inner products and inner product spaces.

Inner products

We have defined distances and norms to measure whether two signals are different from each other and to measure the “size” of a signal. However, it is possible for two pairs of signals with the same norms and distance to exhibit different behavior - an example of this contrast is to pick a pair of orthogonal signals and a pair of non-orthogonal signals, as shown in [link] .

To obtain a new metric that distinguishes between orthogonal and non-orthogonal we use the inner product , which provides us with a new metric of “similarity”.

Definition 1 An inner product for a vector space $\left(X,R,+,·\right)$ is a function $⟨·,·⟩:X×X\to R$ , sometimes denoted $\left(·|·\right)$ , with the following properties: for all $x,y,z\in X$ and $a\in R$ ,

1. $⟨x,y⟩$ = $\overline{⟨y,x⟩}$ (complex conjugate property),
2. $⟨x+y,z⟩$ = $⟨x,z⟩+⟨y,z⟩$ (distributive property),
3. $⟨\alpha x,y⟩$ = $\alpha ⟨x,y⟩$ (scaling property),
4. $⟨x,x⟩\ge 0$ and $⟨x,x⟩=0$ if and only if $x=0$ .

A vector space with an inner product is called an inner product space or a pre-Hilbert space.

It is worth pointing out that properties (2-3) say that the inner product is linear, albeit only on the first input. However, if $R=\mathbb{R}$ , then the properties (2-3) hold for both inputs and the inner product is linear on both inputs.

Just as every norm induces a distance, every inner product induces a norm: ${||x||}_{i}=\sqrt{⟨x,x⟩}$ .

Hilbert spaces

Definition 2 An inner product space that is complete under the metric induced by the induced norm is called a Hilbert space .

Example 1 The following are examples of inner product spaces:

1. $X={\mathbb{R}}^{n}$ with the inner product $〈x,y〉={\sum }_{i=1}^{n}{x}_{i}{y}_{i}={y}^{T}x$ . The corresponding induced norm is given by ${||x||}_{i}=\sqrt{〈x,x〉}=\sqrt{{\sum }_{i=1}^{n}{x}_{i}^{2}}={||x||}_{2}$ , i.e., the ${\ell }_{2}$ norm. Since $\left({\mathbb{R}}^{n},\parallel ·{\parallel }_{2}\right)$ is complete, then it is a Hilbert space.
2. $X=C\left[T\right]$ with inner product $〈x,y〉={\int }_{T}x\left(t\right)y\left(t\right)dt$ . The corresponding induced norm is ${||x||}_{i}=\sqrt{{\int }_{T}x{\left(t\right)}^{2}dt}={||x||}_{2}$ , i.e., the ${L}_{2}$ norm.
3. If we allow for $X=C\left[T\right]$ to be complex-valued, then the inner product is defined by $〈x,y〉={\int }_{T}x\left(t\right)\overline{y\left(t\right)}dt$ , and the corresponding induced norm is ${||x||}_{i}=\sqrt{{\int }_{T}x\left(t\right)\phantom{\rule{3.33333pt}{0ex}}\overline{x\left(t\right)}dt}=\sqrt{{\int }_{T}{|x\left(t\right)|}^{2}dt}={||x||}_{2}$ .
4. $X={\mathbb{C}}^{n}$ with inner product $〈x,y〉={\sum }_{i=1}^{n}{x}_{i}\overline{{y}_{i}}={y}^{H}x$ ; here, ${x}^{H}$ denotes the Hermitian of $x$ . The corresponding induced norm is ${||x||}_{i}=\sqrt{{\sum }_{i=1}^{n}{|{x}_{i}|}^{2}}={||x||}_{2}$ .

Theorem 1 (Cauchy-Schwarz Inequality) Assume $X$ is an inner product space. For each $x,y\in X$ , we have that $|〈x,y〉|\le {||x||}_{i}{||y||}_{i}$ , with equality if ( $i$ ) $y=ax$ for some $a\in R$ ; ( $ii$ ) $x=0$ ; or ( $iii$ ) $y=0$ .

Proof: We consider two separate cases.

• if $y=0$ then $⟨x,y⟩=\overline{⟨y,x⟩}=\overline{⟨0·y,x⟩}=\overline{0}\overline{⟨y,x⟩}=0⟨x,y⟩=0={\parallel x\parallel }_{i}{\parallel y\parallel }_{i}$ . The proof is similar if $x=0$ .
• If $x,y\ne 0$ then $0\le ⟨x-ay,x-ay⟩=⟨x,x⟩-a⟨y,x⟩-\overline{a}⟨x,y⟩+a\overline{a}⟨y,y⟩$ , with equality if $x-ay=0$ , i.e., $x=ay$ for some $a\in R$ . Now set $a=\frac{⟨x,y⟩}{⟨y,y⟩}$ , and so $\overline{a}=\frac{⟨y,x⟩}{⟨y,y⟩}$ . We then have
$\begin{array}{cc}\hfill 0& \le ⟨x,x⟩-\frac{⟨x,y⟩}{⟨y,y⟩}⟨y,x⟩-\frac{⟨y,x⟩}{⟨y,y⟩}⟨x,y⟩+\frac{⟨x,y⟩}{⟨y,y⟩}\frac{⟨y,x⟩}{⟨y,y⟩}⟨y,y⟩\hfill \\ & \le ⟨x,x⟩-\frac{\overline{⟨x,y⟩}⟨x,y⟩}{{||y||}^{2}}={||x||}^{2}-\frac{{|⟨x,y⟩|}^{2}}{{||y||}^{2}}.\hfill \end{array}$
This implies $\frac{{|⟨x,y⟩|}^{2}}{{||y||}^{2}}\le {||x||}^{2}$ , and so since all quantities involved are positive we have $|⟨x,y⟩|\le ||x||·||y||$ .

Properties of inner products spaces

In the previous lecture we discussed norms induced by inner products but failed to prove that they are valid norms. Most properties are easy to check; below, we check the triangle inequality for the induced norm.

Lemma 1 If ${\parallel x\parallel }_{i}=\sqrt{⟨x,x⟩}$ , then ${\parallel x+y\parallel }_{i}\le {\parallel x\parallel }_{i}+{\parallel y\parallel }_{i}$ .

From the definition of the induced norm,

$\begin{array}{cc}\hfill {∥x,+,y∥}_{i}^{2}& =〈x,+,y,,,x,+,y〉,\hfill \\ & =〈x,,,x〉+〈x,,,y〉+〈y,,,x〉+〈y,,,y〉,\hfill \\ & ={∥x∥}_{i}^{2}+〈x,,,y〉+\overline{〈x,,,y〉}+{∥y∥}_{i}^{2}\hfill \\ & ={∥x∥}_{i}^{2}+2\mathrm{real}\left(〈x,,,y〉\right)+{∥y∥}_{i}^{2}.\hfill \end{array}$

At this point, we can upper bound the real part of the inner product by its magnitude: $\mathrm{real}\left(〈x,,,y〉\right)\le |〈x,,,y〉|$ . Thus, we obtain

$\begin{array}{cc}\hfill {∥x,+,y∥}_{i}^{2}& \le {∥x∥}_{i}^{2}+2|〈x,,,y〉|+{∥y∥}_{i}^{2},\hfill \\ & \le {∥x∥}_{i}^{2}+2{∥x∥}_{i}{∥y∥}_{i}+{∥y∥}_{i}^{2},\hfill \\ & \le {\left({∥x∥}_{i}+∥{y}_{i}∥\right)}^{2},\hfill \end{array}$

where the second inequality is due to the Cauchy-Schwarz inequality. Thus we have shown that ${∥x,+,y∥}_{i}\le {∥x∥}_{i}+{∥y∥}_{i}$ . Here's an interesting (and easy to prove) fact about inner products:

Lemma 2 If $〈x,,,y〉=0$ for all $x\in X$ then $y=0$ .

Proof: Pick $x=y$ , and so $〈y,,,y〉=0$ . Due to the properties of an inner product, this implies that $y=0$ .

Earlier, we considered whether all distances are induced by norms (and found a counterexample). We can ask the same question here: are all norms induced by inner products? The following theorem helps us check for this property.

Theorem 2 (Parallelogram Law) If a norm $∥·∥$ is induced by an inner product, then ${∥x,+,y∥}^{2}+{∥x,-,y∥}^{2}=2\left({∥x∥}^{2}+{∥y∥}^{2}\right)$ for all $x,y\in X$ .

This theorem allows us to rule out norms that cannot be induced.

Proof: For an induced norm we have ${∥x∥}^{2}=〈x,,,x〉$ . Therefore,

$\begin{array}{cc}\hfill {∥x,+,y∥}^{2}+{∥x,-,y∥}^{2}& =〈x,+,y,,,x,+,y〉+〈x,-,y,,,x,-,y〉,\hfill \\ & =〈x,,,x〉+〈x,,,y〉+〈y,,,x〉+〈y,,,y〉+〈x,,,x〉-〈x,,,y〉-〈y,,,x〉+〈y,,,y〉,\hfill \\ & =2〈x,,,x〉+2〈y,,,y〉,\hfill \\ & =2\left({∥x∥}^{2}+{∥y∥}^{2}\right).\hfill \end{array}$

Example 2 Consider the normed space $\left(C\left[T\right],{L}_{\infty }\right)$ , and recall that ${∥x∥}_{\infty }={sup}_{t\in T}|x\left(t\right)|$ . If this norm is induced, then the Parallelogram law would hold. If not, then we can find a counterexample. In particular, let $T=\left[0,2\pi \right]$ , $x\left(t\right)=1$ , and $y\left(t\right)=cos\left(t\right)$ . Then, we want to check if ${∥x,+,y∥}^{2}+{∥x,-,y∥}^{2}=2\left({∥x∥}^{2}+{∥y∥}^{2}\right)$ . We compute:

$\begin{array}{cc}\hfill {∥x∥}_{\infty }& =1,\hfill \\ \hfill {∥y∥}_{\infty }& =1,\hfill \\ \hfill {∥x,+,y∥}_{\infty }& =∥1,+,cos,\left(,t,\right)∥=\underset{t\in T}{sup}|1+cos\left(t\right)|=1+1=2,\hfill \\ \hfill {∥x,-,y∥}_{\infty }& =∥1,-,cos,\left(,t,\right)∥=\underset{t\in T}{sup}|1-cos\left(t\right)|=1-\left(-1\right)=2.\hfill \end{array}$

Plugging into the two sides of the Parallelogram law,

$\begin{array}{cc}\hfill {2}^{2}+{2}^{2}& =2\left({1}^{2}+{1}^{2}\right),\hfill \\ \hfill 8& =4,\hfill \end{array}$

and the Parallelogram law does not hold. Thus, the ${L}_{\infty }$ norm is not an induced norm.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
why its coecients must have a power-law rate of decay with q > 1/p. ?