# Tagged Questions

For question involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.

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### Can the composite of two projections really fail to be a projection?

Let $H$ denote a Hilbert space. For any closed subspace $C \subseteq H$, write $P_C$ for the orthogonal projection onto $C$. Then according to wikipedia, the composite $P_U \circ P_V$ needn't be a ...
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### Homeomorphisms between infinite-dimensional Banach spaces and their spheres

As I know Cz. Bessaga has proved that an infinite-dimensional Banach space is homeomorphic to its unit sphere. Unfortunately I do not have his book but I want to know is this theorem true without ...
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### What is a Hilbert space?

I've just seen a question about Hilbert Subspaces. This made me wonder what a Hilbert space is. Can anyone explain in layman's terms?
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### Norm inequality for sum and difference of positive-definite matrices

If $X_{1}$ and $X_{2}$ are positive definite matrices, how to show that $\left\Vert X_{1}-X_{2}\right\Vert \le\left\Vert X_{1}+X_{2}\right\Vert$ for the spectral norm? and how about for the nuclear ...
Suppose that $\ell^2 = \biggl\{(x_n)_n \in \mathbb{K}^{\mathbb{N}_0} \biggm| \sum_{n=1}^{\infty}|{x_n}^2| < +\infty \biggr\}$ is a Hilbert-space with the inproduct $\langle\cdot,\cdot\rangle_2: \... 2answers 313 views ### Show that linear Operator on$\ell^2$is unbounded Currently, I am preparing for a next semester course and trying to figure out some basic concepts in functional analysis. Let$T:\mathcal{D}(T)\to \ell^2$be defined by $$T((x_n)_{n\in\mathbb{N}})=(... 3answers 1k views ### Showing that the orthogonal projection in a Hilbert space is compact iff the subspace is finite dimensional Suppose that we have a Hilbert Space H and M is a closed subspace of H. Let T\colon H\rightarrow M be the orthogonal projection onto M. I have to show that T is compact iff M is finite ... 1answer 426 views ### Uniform mean ergodic theorem I'm working in Einsiedler and Ward's book on Ergodic Theory and in Exercise 2.5.4 they want to prove the following$$\lim_{N - M \to \infty} \frac{1}{N - M} \sum_{n = M}^{N - 1} U_T^n f \to P_T f.$$... 1answer 73 views ### Does proper contraction on Hilbert space necessarily lead to convergence in norm to zero? I was asked this in functional analysis class: Let \mathbb{H} be a Hilbert space and we are given an operator T satisfying: || Th || < ||h|| for all h \in H . We are asked if ... 1answer 167 views ### Hilbert space, dense, orthogonal complement. Suppose H is a Hilbert space, A and B are two subspaces. A is closed and B is dense. If A^\perp \cap B=\{0\}, or in other words, \forall b\in B, the projection to A is not 0, can we ... 1answer 279 views ### Dense subspace of L^{2}[0,1] I know that C[0,1] is dense in L^{2}[0,1] but is \{f\in C^{2}[0,1]:f(0)=f(1)=0\} dense in L^{2}[0,1]? 2answers 164 views ### Norm of a 2\times 2 matrix as a Hilbert space operator Work in the Hilbert space \mathbb C^2. Let$$A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$be a matrix with entries in \mathbb C, and let A ... 1answer 954 views ### Direct sum of orthogonal subspaces I'm working on the following problem set. Let \mathcal{H} be a Hilbert space and A and B orthogonal subspaces of \mathcal{H}. Prove or disprove: 1) A \oplus B is closed, then A and B ... 1answer 278 views ### Spectrum proofs Let T be a densely defined closed unbounded operator on a Hilbert space H. Show that if \lambda is a point in the residual spectrum of T, then \bar{\lambda} is in the point spectrum of the ... 2answers 2k views ### Tensor product of operators We know that if T_1 is a linear bounded operator on a Hilbert space H_1 and T_2 is a linear bounded operator on a Hilbert space H_2 there exists a unique linear bounded operator T on H_1 \... 2answers 265 views ### Is duality an exact functor on Banach spaces or Hilbert spaces? Let V,V',V'' and W be vector spaces over k. Then, it is known that \operatorname{Hom}(\cdot,V) is a contravariant exact functor, i.e. for each exact sequence 0\to V'\to V\to V'' \to 0, and ... 2answers 314 views ### Unit Ball of \mathcal{l}_2 Let B(\mathcal{l}_2) :=\{x \in \mathcal{l}_2 : \|x \| \leq 1 \} and S(\mathcal{l}_2) :=\{x \in \mathcal{l}_2 : \|x \| = 1 \} be the unit ball and the unit sphere of \mathcal{l}_2, respectively. ... 2answers 2k views ### C[0,1] is not Hilbert space Prove that the space C[0,1] of continuous functions from [0,1] to \mathbb{R} with the inner product \langle f,g \rangle =\int_{0}^{1} f(t)g(t)dt \quad is not Hilbert space. I know that I ... 2answers 50 views ### For a normal operator is it true that \|T^*T^2\| = \|T^3\|? For a normal operator is it always true that \|T^*T^2\| = \|T^3\|? See the accepted answer for the case in a Hilbert space Update: how about \|T^*T^2\| = \|T\|^3 in a Hilbert space 1answer 139 views ### Explicit characterization of dual of H^1 Let's start by some well-known facts: H^1(\mathbb{R}) is a Hilbert space, hence there holds the Riesz representation theorem, stating that any linear functional on it can be represented as L = \... 4answers 116 views ### Is every projection on a Hilbert space orthogonal? I'm highly doubtful that the answer is "yes," but I fail to see what's incorrect about this very basic proof I've thought of. If someone could point out my error, I'd appreciate it. My logic is as ... 2answers 419 views ### Optimization problem using Reproducing Kernel Hilbert Spaces I am encountering a problem concerning Reproducing Kernel Hilbert Spaces (RKHS) in the context of machine learning using Support Vector Machines (SVMs). With refernece to this paper [Olivier Chapelle,... 1answer 88 views ### Show that \|e^{tA}\| \le e^{t\|\Re (A)\|} Let X be a complex Hilbert space, and let A be a bounded linear operator on X. Define the real part of A to be \Re(A)=\frac{1}{2}(A^{\star}+A), and define e^{tA}=\sum_{n=0}^{\infty}\frac{1}{... 2answers 1k views ### Spectrum of an Orthogonal Projection Operator I want to show that \sigma(p) = \{ 0,1 \} for any orthogonal projection operator p \notin \{ 0,I \} on a Hilbert space \mathcal{H} . Recall that an orthogonal projection operator p on ... 1answer 1k views ### How to prove this integral operator is compact? T_kf=\int K(x,y)f(y)dy where K(x,y)=\frac{\phi(x)\phi(y)}{|x-y|^{n-\alpha}} \phi(x) is a smooth function on a compact support. f is defined on R^n and K is defined on R^n\times R^n ... 2answers 232 views ### Orthogonality checking in Kreyszig exercise Let H be inner product space with inner product \langle\cdot,\cdot\rangle and norm \lVert \cdot\rVert. Let x,y \in H. Would you help me to prove that \langle x,y\rangle=0 if and only if \... 1answer 611 views ### Exhibiting open covers with no finite subcovers. How do I exhibit an open cover of the closed unit ball of the following: (a) X = \ell^2 (b) X=C[0,1] (c) X= L^2[0,1] that has no finite subcover? 1answer 363 views ### Bounded operator from a Hilbert space to \ell^1 is compact Let H be any Hilbert space. How can we prove that any bounded linear operator T\colon H \to \ell^1 is compact? If we use the fact that the space \ell^1 has Schur property (norm and weak ... 1answer 469 views ### Direct sum \Rightarrow Direct Integral, Tensor product \Rightarrow? Is there a way to define a tensor product over a measure space(=index set) with a continuous measure for Hilbert spaces? For the sum we have the notion of a direct integral, here. 1answer 176 views ### Addition of Unbounded Operators Let H be a (separable complex) Hilbert space and let A and B be two densely-defined, maximally-defined linear operators on H with domains D(A) and D(B) respectively. (By maximall-defined, ... 1answer 347 views ### Is this functional weakly continuous? Take a C^1 function G \colon \mathbb{R}\to \mathbb{R} and define a functional$$\mathcal{G}(u)=\int_0^1G(u(t))\, dt, \quad u \in H^1(0, 1).$$We then have \mathcal{G}\in C^1\big(H^1(0, 1)\to \... 1answer 88 views ### Inequivalent Hilbert norms on given vector space Suppose we have a vector space X. Let \|\cdot\|_1 and \|\cdot\|_2 be two different complete norms on X s.th. X equipped with \|\cdot\|_j, \ j\in\{1,2\} is a Hilbert space. Are there ... 2answers 119 views ### Quantum Mechanics state space In Quantum Mechanics one often deals with wavefunctions of particles. In that case, it is natural to consider as the space of states the space L^2(\mathbb{R}^3). On the other hand, on the book I'm ... 1answer 103 views ### Why do dagger categories supposedly capture the structure of a Hilbert space? A dagger functor is a contravariant endofunctor (\;)^\dagger satisfying X^\dagger = X on objects and f^{\dagger\dagger} on morphisms. It is supposed to model adjoint maps on Hilbert spaces, and ... 1answer 226 views ### Proof involving strongly continuous semigroups. Let (T(t))_{t \geq 0} be a C_{0} -semigroup on a Hilbert space X with an infinitesimal generator A , and let \rho \in (0,1) . I want to prove that \displaystyle \sup_{t \geq 0} \| ... 1answer 176 views ### Can Fourier transform be seen as a decomposition over a basis in a space of tempered distributions Fourier series of a function that belongs to L^2([0,T]) can be seen as a decomposition of this function over an (orthonormal) basis in the Hilbert space L^2([0,T]). Fourier transform of a ... 3answers 205 views ### Gram-Schmidt for uncountable sets? I know that Gram Schmidt can be applied to countable linear independent sets on Hilbert spaces, but what happens if we apply it on uncountable sets? Obviously this process has to fail then (at least ... 1answer 556 views ### The sup norm on C[0,1] is not equivalent to another one, induced by some inner product Let \mathrm{C}[0,1] be the space of continuous functions [0,1]\rightarrow \mathbb{R} endowed with the norm ||x||_{\infty}=\mathrm{max}_{t\in [0,1]}|x(t)|. It is easy to verify that this norm is ... 1answer 787 views ### Showing the basis of a Hilbert Space have the same cardinality I am trying to show that if we have two orthonormal families \{a_i\}_{i\in K} and \{b_j\}_{j\in S} and these are the basis of some Hilbert Space H, then they have the same cardinality. So If I ... 2answers 2k views ### Every Hilbert space has an orthonomal basis - using Zorn's Lemma The problem is to prove that every Hilbert space has a orthonormal basis. We are given Zorn's Lemma, which is taken as an axiom of set theory: Lemma If X is a nonempty partially ordered set with the ... 2answers 2k views ### Inner product on the tensor product of Hilbert spaces Let H_1 and H_2 be Hilbert spaces with inner products \langle\cdot,\cdot\rangle_1 and \langle\cdot,\cdot\rangle_2, respectively. Then H_1\otimes H_2 is at least a pre-Hilbert space (we are ... 1answer 170 views ### Generalized Fourier series in L^2 that do not converge pointwise a.e. For a Hilbert space L^2 we have the notion of an orthonormal basis \{f_j\} being a sequence of orthonormal elements such that any element f in L^2 can be approximated by partial sums in terms ... 1answer 364 views ### Continuous linear image of closed, bounded, and convex set of a Hilbert Space is compact Is my proof of this proposition correct ? And is this proposition well known? Proposition: Let C be a closed, bounded, and convex set in a separable Hilbert space H. Let L : H \to \mathbb{R}^n ... 1answer 423 views ### Show that a subspace of l2 is not complete I would like to know if this exercise is correct. Let \Bbb R^\infty=\{x:\Bbb N\rightarrow \Bbb R: \exists n \text{ such that}\quad x(k)=0 \quad \forall k\geq n\}. Show that (\Bbb R^\infty, \| \... 1answer 113 views ### Strong convergence of an “averaging” operator Let X be an Hilbert space and S:X \rightarrow X be a bounded linear operator with ||S||=1 Define$$T_n= \frac{1}{n} \sum_{r=0}^{n-1} S^r$$I want to show it converges strongly to some ... 1answer 872 views ### Prove or disprove existence of a sequence converging weakly to$0$in an infinite dim Hilbert space This is a problem on an old analysis qual, the prompt is: "Prove or give a counter example: if$H$is an infinite dimensional Hilbert space and$0$is the zero vector in$H$, then there exists a ... 1answer 546 views ### Bounded operator and Compactness problem Let$H$be a Hilbert space with orthonormal basis$(e_{n})_{n\in\mathbb{N}}$. Furthermore, let$T\colon H\rightarrow C[a,b]$be a bounded operator. a) Let$x\in [a,b]$. Show that there is a ... 1answer 73 views ### Space of Jordan curves The space of square-integrable functions$f:[0,1]\rightarrow\mathbb{R}$is well conceivable: it's essentially an$\infty$-dimensional Euclidean space (the Hilbert space$L^2$) with well interpretable ... 1answer 1k views ### How to find an orthonormal basis for$L^2(\mathbb{R},\mathbb{C})$? Consider the Hilbert space$X:=L^2(\mathbb{R},\mathbb{C})$Now consider the operator that takes the second derivative, i.e.$A := \partial_{x}^2$, i.e.$A: H^2(\mathbb{R},\mathbb{C}) \...
Show that if $P$ and $Q$ are two orthogonal projections with orthogonal ranges, then $P+Q$ is also an orthogonal projection. First I need to show $(P+Q)^\ast = P+Q$. I am thinking that since \begin{...