# 1.6 Hilbert spaces

 Page 1 / 1
This module will provide an introduction to the concepts of Hilbert spaces.

## Hilbert spaces

A vector space $S$ with a valid inner product defined on it is called an inner product space , which is also a normed linear space . A Hilbert space is an inner product space that is complete with respect to the norm defined using the innerproduct. Hilbert spaces are named after David Hilbert , who developed this idea through his studies of integral equations. We define our valid norm using the innerproduct as:

$(x)=\sqrt{x\dot x}$
Hilbert spaces are useful in studying and generalizing theconcepts of Fourier expansion, Fourier transforms, and are very important to the study of quantum mechanics. Hilbert spacesare studied under the functional analysis branch of mathematics.

## Examples of hilbert spaces

Below we will list a few examples of Hilbert spaces . You can verify that these are valid inner products at home.

• For $\mathbb{C}^{n}$ , $x\dot y=y^Tx=\begin{pmatrix}\overline{{y}_{0}} & \overline{{y}_{1}} & \dots & \overline{{y}_{n-1}}\\ \end{pmatrix}\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\\ ⋮\\ {x}_{n-1}\end{array}\right)=\sum_{i=0}^{n-1} {x}_{i}\overline{{y}_{i}}$
• Space of finite energy complex functions: ${L}^{2}(\mathbb{R})$ $f\dot g=\int \,d t$ f t g t
• Space of square-summable sequences: ${\ell }^{2}(\mathbb{Z})$ $x\dot y=\sum$ x i y i

a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
how did I we'll learn this
f(x)= 2|x+5| find f(-6)
f(n)= 2n + 1
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Got questions? Join the online conversation and get instant answers!