# 15.6 Common hilbert spaces

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This module will give an overview of the most common Hilbert spaces and their basic properties.

## Common hilbert spaces

Below we will look at the four most common Hilbert spaces that you will have to deal with when discussing and manipulatingsignals and systems.

$\mathbb{R}^{n}$ (reals scalars) and $\mathbb{C}^{n}$ (complex scalars), also called ${\ell }^{2}(\left[0 , n-1\right])$

$x=\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\\ \dots \\ {x}_{n-1}\end{array}\right)$ is a list of numbers (finite sequence). The inner product for our two spaces are as follows:

• Standard inner product $\mathbb{R}^{n}$ :
$x\dot y=y^Tx=\sum_{i=0}^{n-1} {x}_{i}{y}_{i}$
• Standard inner product $\mathbb{C}^{n}$ :
$x\dot y=\overline{y^T}x=\sum_{i=0}^{n-1} {x}_{i}\overline{{y}_{i}}$

Model for: Discrete time signals on the interval $\left[0 , n-1\right]$ or periodic (with period $n$ ) discrete time signals. $\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\\ \dots \\ {x}_{n-1}\end{array}\right)$

$f\in {L}^{2}(\left[a , b\right]())$ is a finite energy function on $\left[a , b\right]$

## Inner product

$f\dot g=\int_{a}^{b} f(t)\overline{g(t)}\,d t$
Model for: continuous time signals on the interval $\left[a , b\right]$ or periodic (with period $T=b-a$ ) continuous time signals

$x\in {\ell }^{2}(\mathbb{Z})$ is an infinite sequence of numbers that's square-summable

## Inner product

$x\dot y=\sum_{i=()}$ x i y i
Model for: discrete time, non-periodic signals

$f\in {L}^{2}(\mathbb{R})$ is a finite energy function on all of $\mathbb{R}$ .

## Inner product

$f\dot g=\int_{()} \,d t$ f t g t
Model for: continuous time, non-periodic signals

## Associated fourier analysis

Each of these 4 Hilbert spaces has a type of Fourier analysis associated with it.

• ${L}^{2}(\left[a , b\right])$ → Fourier series
• ${\ell }^{2}(\left[0 , n-1\right]())$ → Discrete Fourier Transform
• ${L}^{2}(\mathbb{R})$ → Fourier Transform
• ${\ell }^{2}(\mathbb{Z})$ → Discrete Time Fourier Transform
But all 4 of these are based on the same principles (Hilbert space).
Not all normed spaces are Hilbert spaces
For example: ${L}^{1}\left(ℝ\right)$ , $(, f)=\int \left|f(t)\right|\,d t$ . Try as you might, you can't find an inner product thatinduces this norm, i.e. a $·\dot ·$ such that
$f\dot f=\int \left|f(t)\right|^{2}\,d t^{2}=(, f)^{2}$
In fact, of all the ${L}^{p}(\mathbb{R})$ spaces, ${L}^{2}(\mathbb{R})$ is the only one that is a Hilbert space.

Hilbert spaces are by far the nicest. If you use or study orthonormal basis expansion then you will start to see why this is true.

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nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
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da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
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Application of nanotechnology in medicine
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I only see partial conversation and what's the question here!
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please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
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LITNING
scanning tunneling microscope
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Rafiq
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Mahi
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Rafiq
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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write examples of Nano molecule?
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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
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what king of growth are you checking .?
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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Kyle
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
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