# Tagged Questions

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

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### If $\sum (a_n)^2$ converges and $\sum (b_n)^2$ converges, does $\sum (a_n)(b_n)$ converge?

Could someone help me to solve this or at least give me a hint?, I have tried using Cauchy's criterion, the Dirichlet test for convergence, etc, but I can´t prove it.Honestly I don´t know where to ...
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### Graph of symmetric linear map is closed

A homework problem: Let $H$ be a Hilbert space. Let $T:H\rightarrow H$ be a symmetric linear map ($\langle Tx,y\rangle=\langle x,Ty\rangle$). Show that $S$ is bounded. My attempt: I'd ...
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### Meaning of “kernel”

In analysis, there are at least three kinds of "kernel" concepts: In probability theory, there is a concept called transition probability, also called probability kernel, from one measure space $X$ ...
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### Isomorphic Hilbert spaces

As part of a broader proof , I need to show that every two separable Hilbert spaces (that contains a dense countable set) are isomorphic (the linear mapping from one space to the other is injective ...
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### A linear operator on a finite dimensional Hilbert space is continuous

How do I show that a linear function from a Hilbert space $H$ to itself is continuous if $H$ is finite dimensional? Also, what would be an example of a linear function from a Hilbert space to itself ...
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### Are two Hilbert spaces with the same algebraic dimension (their Hamel bases have the same cardinality) isomorphic?

We know that two Hilbert spaces that have orthonormal bases of the same cardinality are isomorphic (as an inner product spaces). My question is: what can we say when we know that their Hamel bases ...
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### 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 ...
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### $f$ an isometry from a hilbert space $H$ to itself such that $f(0)=0$ then $f$ linear.

This question was on an exam and I am not sure how to answer it. I mostly tried writing zero in different ways and tried lots of algebra to get something out. I also tried to use the fact that $H$ is ...
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### 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 ...
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### Showing a certain subspace of Hilbert space is dense

Let H be the Hilbert space of square-summable sequences of reals. A few years ago I thought I had proved that the subspace Z of real sequences with only finitely many nonzero terms, such that they ...
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### Does $S^\bot+T^\bot = (S\cap T)^\bot$ hold in infinite-dimensional spaces?

If $S$ and $T$ are subspaces of some finite-dimensional inner product space then $$S^\bot+T^\bot = (S\cap T)^\bot.$$ See, for example, this post or this post Does it hold in infinite-dimensional ...
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I am wondering if there is any condition one can apply (e.g. uniform boundedness?) that ensure the adjoint of a net of SOT-continuous elements is again SOT-continuous? My major question is ...
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### (From Lang $SL_2$) Orthonormal bases for $L^2 (X \times Y)$

Lang $SL_2$ p. 13 :Let $\{\phi_i\}$, $\{\psi_i\}$ be orthonormal bases for $L^2(X)$ and $L^2(Y)$ respectively. Let $$\theta_{ij}(x,y) = \phi_i(x)\psi_i(y).$$ Then $\{\theta_{ij}\}$ is an orthonormal ...
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### I need to show that $K$ is compact and that $co(K)$ is bounded, but not closed.

Let $x_n$ be a sequence in a Hilbert space such that $\left\Vert x_n \right\Vert=1$ and $\langle x_n,\ x_m \rangle =0$, for all $n \neq m$. Let $K= \{ x_n/ n : n \in \mathbb{N} \} \cup \{0\}$. ...
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Let $l^{2}=\left\{x=(x^{(1)},x^{(2)},...):\sum_{i=1}^{\infty }\left\vert x ^{(i)}\right\vert ^{2}<\infty \right\}$. Would you help me to prove that $({\vert|x_n |\vert})$ is bounded sequence and $(... 1answer 84 views ###$l^p$space not having inner product I know that$l^2$space is a Hilbert space. But for other$l^p$spaces, where$p\geq1$, I have to show that they do not satisfy the parallelogram equality. But, I can't find appropriate sequences ... 1answer 81 views ### Wave Operators: Calculus Given Hilbert spaces$\mathcal{H}_0$and$\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: $$U_\#(t)^*=U_\#(-t)=U_\#(t)^{-1}... 1answer 102 views ### Spectral Measures: Square Root Isometric Equality Given a Hilbert space \mathcal{H}. Consider a closed operator:$$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^{**}$$Denote for shorthand:$$H:=A^*A:\quad H=H^*$$Regard elements: ... 0answers 37 views ### Lemma 3.3-7 and Theorem 3.6-2 in Kreyszig's “Introductory Functional Analysis With Applications”: What if completeness is lost? [duplicate] Let X be an inner product space, and let M be a non-empty subset of X. Then we have the following: (a) If the space of M is dense in X, then M^\perp = \{0 \}, that is, x \in X, x \... 2answers 134 views ### Stone's Theorem Integral: Basic Integral Disclaimer This thread has been renewed: Stone's Theorem Integral: Advanced Integral Problem Given a finite Borel measure \mu and a Hilbert space \mathcal{H}. Consider a strongly continuous ... 1answer 276 views ### Weak limits and subsequences Let S:X \to X be a (nonlinear) map between a Hilbert space X. I want to show that S is weakly continuous, so if x_n \rightharpoonup x, then S(x_n) \rightharpoonup S(x). To do this, I have ... 1answer 43 views ### Wave Operators: Reducibility Given Hilbert spaces \mathcal{H}_0 and \mathcal{H}. Consider Hamiltonians:$$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$Denote their evolutions:$$U_\#(t)^*=U_\#(-t)=U_\#(t)^{-1}... 2answers 111 views ###$T:\ell_2\to \ell_2, T(x_1,x_2,\dotsc)=(x_1,x_2/2,x_3/3,\dotsc)T:\ell_2\to \ell_2, T(x_1,x_2,\dotsc)=(x_1,x_2/2,x_3/3,\dotsc)$I need to know whether it is self adjoint and unitary operator given that$x_i\in\mathbb C$I am not able to do it please tell me how ... 0answers 187 views ### Is the closure of a symmetric operator unique? Let$T:D(T)\rightarrow H$be a densely defined symmetric operator in a Hilbert space H. The closure$\overline T$of$T$is defined as the operator whose graph$G(T)$is the closure of the graph of$...
This thread is Q&A. Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$A:\mathcal{D}A\subseteq\mathcal{H}\to\mathcal{K}$$ Then for the kernel: \mathcal{N}A=...