What is L2: Definition and 46 Discussions

In mathematics, a square-integrable function, also called a quadratically integrable function or



{\displaystyle L^{2}}
function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line




{\displaystyle (-\infty ,+\infty )}
is defined as follows.

One may also speak of quadratic integrability over bounded intervals such as


{\displaystyle [a,b]}



{\displaystyle a\leq b}

An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable. For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part.
The vector space of square integrable functions (with respect to Lebesgue measure) form the Lp space with


{\displaystyle p=2}
. Among the Lp spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notions like angle and orthogonality to be defined. Along with this inner product, the square integrable functions form a Hilbert space, since all of the Lp spaces are complete under their respective p-norms.
Often the term is used not to refer to a specific function, but to equivalence classes of functions that are equal almost everywhere.

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  1. Z

    I Exploring the Impact of Moons on James Webb Telescope at Lagrange Point L2

    The James Webb Telescope is in one of Earths Lagrange points, (I believe it's in L2) How does the moons gravity affect this? Do they have to make course corrections?
  2. jackal123

    Finding Lagrange Point L2: Gravity and Harmonics

    So we are finding the L2 Lagrange point, specifically the distance from the earth, or d in this instance. I have used the equation above and I have come out with 1.5 * 10^9 meters as d, or L2's distance from the earth. Can anyone verify this, is the equation correct and is my final distance...
  3. B

    B Why is the Webb Space Telescope placed at L2?

    Why is the Webb placed at L2 (Lagrange point 2)? I read some articles that said it would be perpetually shaded being on the opposite side of the sun and moon, but that makes no sense when the Webb orbit is so huge? So why is it at L2?
  4. E

    I Force of gravity on JWST while orbiting L2

    When the Webb is at a point on its L2 orbit, (not at L2), what direction is the centrifugal force vector compared to the direction of the combined earth-sun gravity vector on the opposite side? Is the direction of this centrifugal vector ALWAYS parallel to the sun-earth plane? or is it always...
  5. bland

    I Why does Webb orbit L2, is it because of the Moon?

    I was already puzzled by the concept of orbiting a Lagrange point and then I find out it's about the same size orbit as the Moon. I am thinking that if there was no Moon that the Earth and the Sun are far enough away to be treated as points and so that there would be an exact distance further...
  6. V

    The James Webb Space Telescope

    Maybe this is more general discussion, but I am excited / nervous about the upcoming launch of the JWST. https://jwst.nasa.gov/content/webbLaunch/countdown.html I can't wait to see the observations this endeavor will bring!
  7. Buzz Bloom

    B Question re Webb telescope at L2

    The diagram in the article seems to say that the Webb will orbit around the unstable L2 point. At any distance near L2 but not exactly at L2 the Webb will tend to move further from L2, unless Webb has an engine and fuel to maintain it in the special orbit. I was not able to find any description...
  8. M

    MHB Express X in terms of L1, L2 and L3. Circle within Circle.

    I do not know if there is a solution to the problem. With the aid of CAD Software, I get X = 3.5 when L1=7.082, L2=0.684, L3=0.876. I don't know how accurate this is. My intuition says there should be a expression for X but I am yet to solve it. Any ideas?
  9. E

    I Fredholm's alternative & L2 convergence

    Hello everyone, I'm currently going through Strauss "introduction to differential equations" and i can't get around a certain proof that he gives on chapter 11.5 page(327 (2nd edition)).Specifically, the proof refers to a certain version of Fredholm's alternative theorem. Assume that we are...
  10. Johnnyallen

    I Gaia Space Telescope and Lagrangian Point 2

    I'm confused (what else is new) about L2. While watching a video from PBS Digital Spacetime about the latest data drop from Gaia Space Telescope, Matt O'Dowd showed a CGI animation of the telescope leaving Earth then circling/orbiting L2 perpendicular to the Earth/sun plane. I thought that the...
  11. F

    In a circuit, what causes current to go from L1 to L2?

    The photograph above this sentence is a photograph of a simple 2 leg circuit of alternating current with a light switch and a light bulb. I know how alternating current constantly switches direction from line 1 to line 2 and then goes from line 2 to line 1, etc. In the diagram in the above...
  12. M

    A Can the convolution operator be diagonalized using the Fourier transform?

    Hi there, I am also familiar with Hilbert spaces and Functional Analysis and I find your question very interesting. I agree that the Fourier transform is a powerful tool for analyzing LTI systems and diagonalizing the convolution operator. As for your question about whether the same can be...
  13. EthanVandals

    Solving Balancing Beam: m1, L1, m2, L2

    Homework Statement In the picture, m1 is 20kg, and the wedge pushes up with a force of 300 N. If the length L1 is 2 meters, solve for m2 and L2. Homework Equations I'm not sure The Attempt at a Solution The picture shows some sort of beam balancing on a fulcrum that is not centered. On the...
  14. D

    Decide the resulting force (F) per meter on L2

    Homework Statement Three long straight parallell wires that conduct electricity called L1, L2 and L3. The angle at L2 is 90 degrees and the distance between L2 and L1, L3 is 0,50 meters. The current through the wires L1 and L3 is 300 A while the current in L2 is 600 A. Decide the resulting...
  15. Imager

    B Understanding the Lagrange Points: What's Going Wrong with L2 and L3?

    The Wiki article shows 5 Lagrange points. I can “see” how the points L1, L4 and L5 points would be balanced by the gravitation of the two bodies, but not the L2 and L3. For L2 and L3, it looks to me like the combination of the Sun’s and Earth’s gravity increase pull and make less stable. So...
  16. P

    B Does the L2 norm of a vector destroy all directional info?

    Sorry I'm a little rusty with my math and proof logic, and this feels like a dumb question, but oh well! The Euclidian norm of a vector in ℝ3 is \|{v}\| = \sqrt{x^2 + y^2 + z^2} where \|{v}\| \geq 0. I'm trying to show that there is always an infinite number of solutions for arbitrary...
  17. kenyanchemist

    I Is Ψ2Px an Eigenfunction of L2 or Lz in Quantum Mechanics?

    hi, am major new on quantum mechanics. please help me understand. is the real wave function Ψ2Px= [Ψ2p+1 +Ψ2p-1]1/2 an eigen function of L2 or Lz? if so, how is it? and if so kindly explain the values of l and m thanks
  18. RJLiberator

    PDE: Proving that a set is an orthogonal bases for L2

    Homework Statement Show that the set {sin(nx)} from n=1 to n=∞ is orthogonal bases for L^2(0, π). Homework EquationsThe Attempt at a Solution Proof: Let f(x)= sin(nx), consider scalar product in L^2(0, π) (ƒ_n , ƒ_m) = \int_{0}^π ƒ_n (x) ƒ_m (x) \, dx = \int_{0}^π sin(nx)sin(mx) \, dx =...
  19. diegzumillo

    Decompose wave packet into eigenvalues of L2, Lz and k

    Homework Statement The free particle wave packet in question is $$\psi=ce^{-(r/r_0)^2}$$ Homework EquationsThe Attempt at a Solution I've been going through books and class notes but I really have no idea where this came from. I'm thinking that if I can decompose this in plane waves I could...
  20. S

    How to prove that the L2 norm is a non-increasing function of time?

    Homework Statement Homework Equations How can I start the proof? Shall I use the Poincare inequality? The Attempt at a Solution Well, I know that this norm is defined by , but still I don't know how to start constructing the proof?
  21. K

    What equation can I use to find the value of L2?

    Hi there, I have the following thread: 1) m2 < m1 and L2 < L1 2) abs(m2-m1)=abs(L2-L1) the values of m2, m1 and L1 are known, what equation can I use to find the value of L2? thank you
  22. T

    Convergence in Uniform and L2 sense, function interpretation

    Let: gn(x) = 1 in [1/4 - 1/n2 to 1/4 + 1/ n2) for n = odd 1 in [3/4-1/n2 to 3/4 + 1/n2) for n = even 0 elsewhere Show the function converges in the L2 sense but not pointwise. My issue is in how I should use the definition of...
  23. H

    MHB Calculating an integral norm in L2

    If I have the following operator for $H=L^2(0,1)$:$$Tf(s)=\int_0^1 (5s^2t^2+2)(f(t))dt$$ and I wish to calculate $||T||$, how do I go about doing this: I know that in $L^2(0,1)$ we have that relation:$$||T||\leq \left ( \int_0^1\int_0^1 |(5s^2t^2+2)|^2dtds\right )...
  24. P

    Why is L1 norm harder to optimize than L2 norm?

    Hi all, I have a basic optimisation question. I keep reading that L2 norm is easier to optimise than L1 norm. I can see why L2 norm is easy as it will have a closed form solution as it has a derivative everywhere. For the L1 norm, there is derivatiev everywhere except 0, right? Why is this...
  25. T

    Does being in C2 imply being in L2 for a function?

    Having bounded set ##U\subset\mathbb{R}^n## with ##C^1## boundary ##\partial U## and a function ##g\in C^2(\partial U)##, does one automatically have ##g\in L^2(\partial U)##? I don't need a proof/explanation, yes/no answer is sufficient.
  26. B

    Proof that a given subspace of C[−1,1] with L2 norm is closed

    Homework Statement Let H= C[-1,1] with L^2 norm and consider G={f belongs to H| f(1) = 0}. Show that G is a closed subspace of H. Homework Equations L^2 inner product: <f,g>\to \int_{-1}^{1}f(t)\overline{g(t)} dt The Attempt at a Solution I've been trying to prove this for a...
  27. E

    L2 norm for complex valued vector

    Let's say I have a vector (4+2i, 1-i), how do I take an L2 norm? Dont tell me I simply do sqrt(16+4+1+1)..?
  28. D

    L2 norm of complex functions?

    Hi, I want to show: \|f-jg\|^2 = \|f\|^2 - 2 \Im\{<f,g>\} + \|g\|^2 However, as far as I understand, for complex functions <f,g> = \int f g^* dt, right? Therefore: \|f-jg\|^2 = <f-jg, f-jg> = \int (f-jg)(f-jg)^* dt = \int (f-jg)(f+jg) dt = \int f^2 + jfg - jfg + g^2 dt = \|f\|^2 + \|g\|^2...
  29. J

    If sequence {x} is in l2, does x_n<k/n follow?

    Homework Statement Suppose we have a sequence {x} = {x_1, x_2, ...} and we know that \{x\}\in\ell^2, i.e. \sum^\infty x^2_n<\infty. Does it follow that there exists a K>0 such that x_n<K/n for all n? Homework Equations The converse is easy, \sum 1/n^2 = \pi^2/6, so there would be a finite...
  30. O

    What is the history of L1 and L2 cache in microprocessors?

    Hello! According to many sites L1 is internal cache and Level 2 cache, cache memory that is external to the microprocessor. but according to this intel core2duo box (photo) , intel is selling this L2 cache on processor not on another chip. http://i40.tinypic.com/i4fu4n.jpg why is...
  31. I

    Function belongs to L2 and L_infinity, but is not vanishing

    I got this open problem from my advisor: find a time function which belongs to L2 and L_infinity, but is not vanishing as t -> infinity. Lp here is the p-norm space of functions http://en.wikipedia.org/wiki/Lp_space#Lp_spaces_2 I've worked on this for two weeks now, and still not found the way...
  32. M

    Accepted (precise) value for lunar L2 point?

    Does anyone know where I can find one? I've written a program to test the stability of the lunar L2 point by placing a massless test object in orbit there and I've found an optimised value based on the approximation r_{m}(\frac{M_{M}}{3M_{E}})^{1/3} where r_{m} is the radius of the...
  33. C

    Show that l2 space separable

    Homework Statement 1. Prove that if a metric space (X,d) is separable, then (X,d) is second countable.2. Prove that \ell^2 is separable. Homework Equations The Attempt at a Solution 1. \{ x_1,\ldots,x_k,\ldots \} is countable dense subset. Index the basis with rational numbers, \{ B(x,r) | x...
  34. D

    So, my question is, where did I go wrong in my approach?

    Let S be a subspace of L^{2}(\left[0,1\right]) and suppose \left|f(x)\right|\leq K \left\| f \right\| for all f in S. Show that the dimension of S is at most K^{2} --------- The prof hinted us to use Bessel's inequality. Namely, let \left\{ u_1,\dots, u_m \right\} be a set of...
  35. R

    Is There an Inequality Between L1 and L2 Norms?

    Homework Statement \|x\|_2\le\|x\|_1\le\sqrt{n}\|x\|_2 where |x|1 is the l1 norm and |x|2 is the l2 normHomework Equations See aboveThe Attempt at a Solution I have \|\mathbf{x}\|_1 := \sum_{i=1}^{n} |x_i| and \|x\|_2 = \left(\sum_{i\in\mathbb N}|x_i|^2\right)^{\frac12} I have tried to...
  36. mnb96

    L2 Norm of +Infinity: Admitted & Defined

    Hello, I have a (infinite dimensional) vector space and defined an inner product on it. The vectors element are infinite sequence of real numbers (x_1, x_2,\ldots). The inner product has the common form: x_iy_i The problem now is that the vectors have an infinite number of elements, so the...
  37. B

    Integer tuples with equal L1 and L2 norms

    Let x and y be n-tuples of non-negative integers. Furthermore, sum x_i = sum y_i and, sum x_i^2 = sum y_i^2 Is it true that x must be a permutation of y? Cheers!
  38. Y

    Prove sinx/x in L2 but not in L1

    Homework Statement Prove that f(x)=\sin(\pi x)/(\pi x) is in L^2(R) but not in L^1(R) This is in a chapter of the book dealing with Inverse Fourier Transform f is in L^1 if \int|f|<\infty f is in L^2 if \sqrt{\int|f|^2}<\infty Homework Equations I just have no idea how to do it The Attempt...
  39. A

    Eventual boundedness of nth derivative of an analytic function in L2 norm

    I'm trying to show that if f(x) is analytic, then for large enough n, || f^{(n)} (x) || \leq c n! || f(x) ||, where || f ||^2=\int_a^b{|f|^2}dx and f^{(n)} denotes the nth derivative. I tried to use the Taylor series, and then manipulated some inequalities, but I wasn't getting...
  40. J

    Functions in L1 that are not in L2

    It is clear that there are functions in L2 that are not in L1, but what about the other way? And what effect does considering L2(R) versus L2([a,b]) have? Thanks.
  41. A

    Question about L2 which is not L-infinity

    Homework Statement Construct a function u of the space H'(B), where B is a unit sphere in R^3, which does not belong to L∞(B). Homework Equations (relevant facts) all L2 are hilbert, therefore the problem reduces for us finding an L2 function which is not L-infinity. The Attempt...
  42. I

    Calculating L2 Lagrangian Point: A Beginner's Guide

    Hi this is my first time posting on this forum. I have an question about Lagragian points. I was trying to find L2 lagrangian point, a point that lies on the line defined by the two large masses, beyond the smaller of the two. Here, the gravitational forces of the two large masses balance...
  43. T

    Convergence with L2 norm functions

    Homework Statement (I'm posting this because my proofs seem to be lousy. I want to see if I'm missing anything.) Show that if f_n \in L^2(a,b) and f_n \rightarrow f in norm, then <f_n,g> \rightarrow <f,g> for all g \in L^2(a,b) Homework Equations L^2(a,b) is the space of...
  44. D

    L1-, L2-, Linfty-Norm Proofs -

    L1-, L2-, Linfty-Norm Proofs - Please Help! Homework Statement Show that ||x||1 < or = n||x||infinity and ||x||2 < or = sqrt(n)*||x||infinity for x exists in the set of all real numbers. Homework Equations ||x||2 is defined here: http://mathworld.wolfram.com/L2-Norm.html ||x||1 is...
  45. wolram

    L1 L2 Points: Influence & Model Dependence

    Can some one tell me how big an area of influence the lagrange L1, L2 etc points have ? or are they model depandant ? Thanks.
  46. P

    L2 transformation reduces to the L1 transformation

    Hello. I am having trouble answering the following question: "Show that the L2 transformation reduces to the L1 transformation when the two reference frames are in standard configuration." Am I wrong to assume that r = xi + yj + zk Any help would be beautiful! Thanx much