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

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Vector, Hilbert, Banach, Sobolev spaces

Trying to wrap my head around all these different spaces. Which one is the most general? Can you summarize the differences between them? Is there a notable space that I missed?
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1answer
728 views

Different versions of Riesz Theorems

In Wikipedia, there are three versions of Riesz theorems: 1 The Hilbert space representation theorem for the (continuous) dual space of a Hilbert space; 2 The representation theorem for ...
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395 views

Convergence of a sequence of periodic functions

Motivated by the homogenization theory which studies the effects of high-frequency oscillations in the coefficients upon solutions of PDE, I am thinking about the following question. Let the ...
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3answers
181 views

Why is it useful to express PDE solutions as $L^2$-convergent series?

The existence of an $L^2$ orthonormal basis consisting of eigenfunctions of a Sturm-Liouville equation helps us to express the solutions of various ODEs and PDEs as infinite series. However, in the ...
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1answer
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Operator norm and tensor norms

I have a linear operator $A\in\mathcal{L}(X,Y)$ where $X$ and $Y$ are some Banach spaces (or Hilbert spaces would also do, if that simplifies the answer.). The operator norm of $A$ is given by $$ ...
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2answers
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Elegant proof that $L^2([a,b])$ is separable

Is anybody aware of, or can provide at least an outline, of a proof that the Hilbert space of Lebesgue functions square-integrable on the closed real interval [a,b], equipped with the $L^2$ norm, is ...
4
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1answer
705 views

Summation of inner products

I can't seem to find a way of asking a sub-question in relation to does linearity of inner product hold for infinite sum, which is in itself too generic a question for my purposes. Could someone ...
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3answers
723 views

Orthonormal basis in $L^2(\Omega)$

In the one dimension case, where $\Omega\subseteq{\bf R}$ is a bounded domain, for example $\Omega=[0,2\pi]$, one can find the orthonormal basis for $L^2(\Omega)$: $$\{e_n\}_{n\in {\bf Z}}$$ where ...
7
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1answer
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Energy estimate of the differential equation $\dot{x}=Ax$

Conside the differential equation $$\dot{x}=Ax,\qquad x(t):{\bf R}\to{\mathcal H}$$ where $\mathcal{H}$ is a Hilbert space and $A$ is a bounded linear operator. With the initial condition, one can ...
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About the operator theory in Hilbert space

Let $H$ be a hilbert space. $L(H)$ be the set of linear operators on $H$. Suppose that $S,T\in L(H)$ and $S\geq0$, $\|Sx\|=\|Tx\|$ for every $x\in H$. Can I conclude that $S=\sqrt{T^*T}$?
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Given two basis sets for a finite Hilbert space, does an unbiased vector exist?

Let $\{A_n\}$ and $\{B_n\}$ be two bases for an $N$-dimensional Hilbert space. Does there exist a unit vector $V$ such that: $$(V\cdot A_j)\;(A_j\cdot V) = (V\cdot B_j)\;(B_j\cdot V) = 1/N\;\;\; \ ...
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How to prove that square-summable sequences form a Hilbert space?

Let $l^2$ be the set of sequences $x = (x_n)_{n\in\mathbb{N}}$ ($x_n \in \mathbb{C}$) such that $\sum_{k\in\mathbb{N}} \left|x_k\right|^2 < \infty$, how can I prove that $l^2$ is a Hilbert space ...
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2answers
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Hilbert spaces, square integrability etc

(Someone may please change the title if they can think of a better one) We have a Hilbert Space $\mathcal{H}$ that consists of all functions $\psi(x)$ such that $\int_{-\infty}^{\infty} |\psi(x)|^2 ...
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0answers
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Counting balls in Hilbert spaces

Let $W$ be a real Hilbert space of dimension $n$ and $V$ a Hilbert subspace of dimension $m$. Assume that $f_1,\cdots,f_k$ are points in $W$ such that the following holds: there exists $\sigma>0$ ...
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On the isometry between bounded linear operators and the dual of nuclear linear operators

Let $H$ be a separable Hilbert space. Let $(e_i)_i$ be an orthonormal basis. For any bounded linear map $T$ we write, whenever possible $$\operatorname{tr} T := \sum_{i}^{\infty} \langle T e_i, e_i ...
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1answer
202 views

Form of the inner product in $\ l_2$

Is it true that every inner product in $\ l_2 $ is of the form $\langle x,y\rangle_a =\sum_{n=1} ^ {\infty} {a_n x_n y_n}$ ? (Of course $\ x=(x_n) , y=(y_n) $ are in $\ l_2 $ .)
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1answer
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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 $$\lim_{N - M \to \infty} \frac{1}{N - M} \sum_{n = M}^{N - 1} U_T^n f \to P_T f.$$ ...
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1answer
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Convexity in Hilbert Spaces

Let $\mathbb{H}$ be a Hilbert space on which $\Phi$ is a bounded linear functional. Define $f : \mathbb{H} \rightarrow \mathbb{R} : x \mapsto ||x||^2 + \Phi (x)$. Prove that each (nonempty) closed ...
7
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1answer
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Does there exist a real Hilbert space with countably infinite dimension as a vector space over $\mathbb{R}$?

Essentially what the title says - where to me a Hilbert space is a complete (Hermitian) inner product space, am I safe to assume every such real Hilbert space is of uncountable dimension over ...
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3answers
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An orthonormal set cannot be a basis in an infinite dimension vector space?

I'm reading the Algebra book by Knapp and he mentions in passing that an orthonormal set in an infinite dimension vector space is "never large enough" to be a vector-space basis (i.e. that every ...
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3answers
397 views

Soft Question Hilbert Space Geometry

Just a quick question about the geometry of Hilbert spaces from an intuitive standpoint. Maybe just assuming we're working with $L^2$ would simplify the situation. Basically, in something like ...
3
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1answer
436 views

Quadratic minimization in a Hilbert space

If $A$ is a positive definite matrix, then the solution to the minimization problem $(1/2)x^TAx - b^Tx$ is given by $A^{-1}b$. I'm interested in the generalization of this to a Hilbert space. What ...