



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:
$(x)=\sqrt{x\cdot x}$
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
$\mathbb{C}^{n}$ ,
$$x\cdot y=y^Tx=\begin{pmatrix}\overline{{y}_{0}} & \overline{{y}_{1}} & \dots & \overline{{y}_{n1}}\\ \end{pmatrix}\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\\ \vdots \\ {x}_{n1}\end{array}\right)=\sum_{i=0}^{n1} {x}_{i}\overline{{y}_{i}}$$
 Space of finite energy complex functions:
${L}^{2}(\mathbb{R})$
$$f\cdot g=\int \,d t$$∞
∞
f
t
g
t
 Space of squaresummable sequences:
${\ell}^{2}(\mathbb{Z})$
$$x\cdot y=\sum $$∞
∞
x
i
y
i
Questions & Answers
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or infinite solutions?
Kim
if A not equal to 0 and order of A is n prove that adj (adj A = A
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J, combine like terms 7x4y
im not good at math so would this help me
how did I we'll learn this
Need to simplify the expresin. 3/7 (x+y)1/7 (x1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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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.
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Source:
OpenStax, Fundamentals of signal processing. OpenStax CNX. Nov 26, 2012 Download for free at http://cnx.org/content/col10360/1.4
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