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

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On the isomorphism between bounded sesquilinear forms and bounded operators between two Hilbert spaces

Let $H$ and $K$ be two Hilbert spaces. Let $S(H,K)$ be the vector space of bounded sesquilinear forms $u:H\otimes \overline{K}\to\mathbb{C}$, and let $B(H,K)$ be bounded linear operators from $H$ to ...
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45 views

Confusion related to hilbert space

I was reading this article related to Hilbert spaces I didn't get why the first function space is not Hilbert space. I mean I can define the same norm $||f|| =\max_{a\leqslant x\leqslant b} ...
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57 views

Why is $L^2$ function Hilbert space not defined for Riemann Integral

The space of square Lebesgue integrable functions is said to be a Hilbert space. Why is if the integral is Riemann then this is not a Hilbert space? In other words, why not the space of Riemann square ...
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28 views

homomorphism question

Let B(H) be the set of bounded linear operators on a hilbert space H. Let F be a unital commutative subspace of B(H). Give an example of a homomorphism h from F to the complex numbers such that h is ...
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55 views

Surjective homomorphism example

What is an example of a surjective homomorphism $B(H)\to\mathbb C$, where $B(H)$ is the set of bounded linear operators on a Hilbert space $H$, and $\mathbb C$ is the complex numbers.
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30 views

Zero Operators on Complex Hilbert Space

This is a problem from Kreyszig's Introdcutory Functional Analysis with Applications. If for any $x$ in a complex Hilbert Space $<Tx, x> = 0$, show that $T\equiv 0$. Any clue?
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29 views

A couple of proofs on a spectrum

Let $T$ be a normal bounded operator. Let ${\lambda}$ be in $({\sigma}(T))$. Without invoking general algebra theories, show that: a) $p({\lambda},{\lambda}^*)$ is in $({\sigma}(T))$ for all ...
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44 views

Gelfand transform explicity

Let $T$ be a bounded normal operator. Let $A$ be the algebra generated by $T$ and $T^*$. What is the explicit Gelfand transform $G:A\to C(\sigma(T))$? My book says the image of $T$ is the ...
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54 views

Expression for orthogonal projection onto Hilbert space (is related to Galerkin method)

Let $H=L^2(\Omega)$ and $V=H^1(\Omega)$. Suppose that $\{v_j\}$ is a basis for $H$ and $V$ (not necessarily orthogonal). Let $V_m = \text{span}(v_1, ..., v_m)$. Define a projection operator $P_m:H ...
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46 views

Hahn-Banach separation theorem for Hilbert spaces

What is the strongest form of the Hahn-Banach separation theorem for Hilbert spaces? Could you please provide a reference?
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77 views

Does this statement about Hilbert spaces make any sense?

I have found this tweet about git and don't know what to make of it. I think it's written as a joke, but it could have been written in Chinese, and I'd understand ...
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52 views

Difference between unconditional and absolute convergence in Banach spaces

One can show that in any finite-dimensional normed vector space absolute convergence is equivalent to unconditional convergence. It's not hard to show that if we have an orthonormal sequence in ...
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40 views

Direct product of a family of Hilbert spaces

Let $\{H_i\}_{i\in I}$ be a family of Hilbert spaces, defined $$H = \{f\in \Pi_{i\in I} H_i, \sum_{i\in I}|f(i)|^2<\infty\}$$ and inner product $\langle f,g\rangle : = \sum_{i\in I} \langle ...
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46 views

Reproducing kernel Hilbert sapce

I encountered the following claim (verbatim): Theorem Let $V$ be a subspace of $L^2(\mathbb{R})$ and $\{e_n\}$ be a orthonormal basis of $V$. The $V$ is a reproducing kernel Hilbert space with kernel ...
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1answer
45 views

In a dagger category, is there a name for morphisms $f : X \rightarrow Y$ with $\mathrm{id}_X = f^\dagger \circ f$?

In a dagger category, is there a name for morphisms $f : X \rightarrow Y$ with $\mathrm{id}_X = f^\dagger \circ f$? Clearly, every such arrow is a split monomorphism; further, if such an $f$ is ...
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32 views

Gelfand Transform in a specific case

What is the gelfand transform of an operator in the algebra generated by a bounded normal operator and it's adjoint? Thanks
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35 views

Unbounded extension of bounded operator

Is it possible to construct an unbounded extension of bounded densely defined operator? To be more concrete, let $\mathcal{H}$ be Hilbert space, $\mathcal{D}\subset\mathcal{H}$ - a dense subset, ...
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2answers
54 views

Parseval's identity, decomposition of inproduct.

Hoi, if $H$ is a seperable real Hilbertspace and $(e_n)$ orthonormal basis, then Parseval's identity $$\sum_n\left\langle x,e_n \right\rangle^2 = \left\|x\right\|^2 = \left\langle x,x ...
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41 views

Operators on a Hilbert space question

For a Borel measure $\mu$ define $\langle S_\mu x,y\rangle=\int_H\langle x,z\rangle \langle y,z\rangle \mu(z)$. An exercise in my book that I am reading says that I could find a $\mu$ s.t. $S_\mu$ ...
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42 views

Dense subspace in a Hilbert space

Let $H$ be a Hilbert Space and $\{e_n\}_{n\in\mathbb{N}}$ an orthonormal basis. Now let $(x_n)$ be a sequence in $H$ satisfying $$\sum_{n=1}^{\infty}||x_n-e_n||^2<1.$$ Prove that ...
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1answer
55 views

Common orthogonal basis for $L^2$ and $H^1$

How can we obtain a common orthogonal basis for the space $L^2(U)$ and $H^1(U)$ for some bounded open subset of $\mathbb{R}^n$? That this can be done is mentioned in Evans's Partial Differential ...
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15 views

characterisation up to unitary equivalence

My book says that the spectral theorem for compact normal operators characterises compact normal operators up to unitary equivalence. It doesn't expand on this so I was wondering what does this mean ...
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25 views

The eigenspace of 0 problem

In the spectral theorem for a compact normal operator, do we exclude the eigenspace corresponding to 0 (assuming its an eigenvalue) from the space decomposition. My reason for asking is this: Every ...
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1answer
36 views

Complex Projective Line

How can I go about showing that a collection of all states is the complex projective line $CP^1$? All I understand at the moment is that an element in $CP^1$ is of the form ...
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45 views

Closed affine subspace of $\mathcal{L}^{2}$

Consider a Hilbert space $\mathcal{L}^{2}=\lbrace X: X-\text{real-valued random variable}, \mathbb{E}(X^{2})<\infty \rbrace$ with the inner product $\langle X,Y\rangle=\mathbb{E}(XY)$. Let ...
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2answers
48 views

Show that $\ell^2(A)$ and $\ell^2(B)$ are isomorphic iff $A$ and $B$ have the same cardinality

Let $A,B$ be sets. Show that $\ell^2(A)$ and $\ell^2(B)$ are isomorphic iff $A$ and $B$ have the same cardinality. (Here $\ell^2(A)$ is the square integrable functions that stand on $A$ with the ...
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37 views

Adjoint of an operator question.

Let T be a normal operator. Prove that $\|T\|^{2n}=\|TT^*\|^n$ Has it got something to do with $\|T\|=\|T^*\|$?
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38 views

weak derivatives of exp(-|x|) and Hilbert Spaces

To which Hilbert Space (W^m,2) of R does the function exp(-|x|) belongs? I know its weak derivative is (-exp(-x) for x>0, exp(x) for x<0 and c0 (arbitrary) for x = 0). This weak derivative is in ...
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31 views

induction on powers of a norm

Let T be a self adjoint operator on a hilbert space. I wish to prove by induction that $||T^n||$=$||T||^n$. I have proved it for n=1 and n=2. So assume it is true for some n Then, ...
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1answer
50 views

Show that H$(I)$ is a closed subspace of $L^2(I)$

EDIT: Original statement is not true, added condition. Let $I$ be the unit interval, define $H(I) = \{f\in AC(I)$ and $f'\in L^2(I)\}$. I want to show that $H(I)$ a closed subspace of $L^2(I)$. ...
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23 views

Palais-Smale condition

I have this function: $$\tilde{f}(x)=f(x)+p(||x||)(x_0,x)$$ where $p\in C^2([0,\infty),\mathbb{R}) $ satisfy $0\leq p\leq 1,|p'(t)|\leq \frac{4}{\delta},$ and $$ p(t)= \begin{cases} 1& t\in ...
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155 views

Is a closure of subspace N and and orthogonal complement of this subspace N orthogonal?

Ok, there is something I do not understand about what I run into today in an online document. I know it might sound simple but I am so new to topology so I am having hard time to understand. As we ...
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1answer
43 views

Counterexample of minimum principle in hilbert space on non closed but convex subspace

As I mentioned at title, I make tiny counterexample for minimum principle. Let $K=C([0,\frac{1}{2}]) \subset H=L^{2}([0,1])$. Then $K$ is convex since every $f,g \in K$, $(\alpha ...
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59 views

Unconditionally convergence in Hilbert space

Let $H$ be a complex Hilbert space and $e_1, e_2,...$ be a countable orthonormal system in $H$, $c_1, c_2,...$ is a sequence of complex numbers. How to prove that if $c_n$ is square-summable then ...
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65 views

What's this standard duality argument?

I'm reading a proof of the Strichartz inequalities. It shows that $$ \| \int_\mathbb{R} e^{-is\Delta}F(s) \, ds \|_{L^2_x} \lesssim \|F\|_{L^{q'}_t L^{r'}_x}, $$ and then says that by duality, $$ ...
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1answer
56 views

On the completeness of inner product spaces.

Let $H$ be a Hilbert space, equipped with an inner product $(\cdot,\cdot)_1$ and norm $\|\cdot\|_1$ induced by it. Let $(\cdot,\cdot)_2$ be other inner product on $H$ and $\|\cdot\|_2$ the norm ...
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49 views

Condition for vector to be in the domain of unbounded operator.

Let $P$ be unbounded self-adjoint operator on some Hilbert space $\mathcal{H}$. We assume that the limit $$ \lim_{\epsilon \searrow 0} \|\exp(-\epsilon^2 P^2/2) P\psi\| $$ exists and is finite. Does ...
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85 views

Ultraweak topology on Banach spaces

If $X$ and $Y$ are Banach spaces with $Y$ reflexive, then the space $\mathcal{B}(X,Y)$ of bounded operators from $X$ to $Y$ is the dual of the projective tensor product of $X$ and $Y^{*}$. As in the ...
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2answers
81 views

Spectral Theorem for normal operators

I want to prove this in the infinite dimensional Hilbert space case. What is the easiest way to go about this (What do I need to know, what theorems do I need,etc). My aim is to show every normal ...
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33 views

Question id a derivative on a Hilbert space

On a Hilbert space $H$; i have this function: $\tilde{f}(x)=f(x)+p(||x||)(x_0,x)$ where $x_0\in H, p\in C^2([0,\infty),\mathbb{R}),f\in C^2(H,\mathbb{R})$ i want to caculate $\tilde{f}', ...
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41 views

Continuity domain for momentum operator

The momentum operator in one dimension in quantum mechanics is $P=-i\frac{d}{dx}$ (with $\hbar=1$). Consider it as an operator on $L_2(0,2\pi)$, the space of square-integrable functions on $(0,2\pi)$. ...
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2answers
78 views

Derivative on Hilbert space

Please, on a Hilbert space what is the derivative of $\displaystyle\frac{x}{||x||}$ ? I know that it's equal to $\displaystyle \frac{1}{||x||}-\frac{\langle x,\cdot\rangle}{||x||^3} x$ but can I ...
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69 views

Supremum calculation

Calculate $\sup(\sum_{k=n+1}^{\infty}\frac{|x_{k}|^{2}}{4^{k} })$, where $x=(x_{1},x_{2},....)$ is a member of $l_{2}$ and the supremum is take over all $x$ with $||x||= 1$. My intuition says the ...
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1answer
45 views

Weak derivative of one parameter group and the domain of its generator

Let $U(t)=\exp(i t A)$ be a one parameter group generated by self-adjoint (unbounded) operator A. It is well-known that if $$ \lim_{t\rightarrow 0} \frac{U(t)\psi-\psi}{t} $$ exists then $\psi$ ...
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1answer
29 views

Distance between Unilateral shift and invertible operators.

I want to prove that the distance between unilateral shift and normal operators is $1$. But I need to prove that $d(S,\operatorname{Inv}(L(H))= 1$, where $H$ is a Hilbert space. Does anyone have any ...
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1answer
80 views

How to find adjoint operator?

Let $(X,\langle\cdot,\cdot\rangle)$ be a Hilbert Space over $K$ with orthonormal basis $(x_n)$, and let $(\lambda_n)\in K$ be a bounded sequence. The mapping $T:X\to X$ is defined by ...
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1answer
64 views

Generalized functions as integral kernels on Hilbert spaces

I'm a physics student and I'm studying functional analysis. I've got a doubt about some operators defined by integral kernels that are generalized functions. Let $L_2(a,b)$ be the Hilbert space of ...
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1answer
23 views

Logical meaning to morphisms between prehilbertian spaces

I was wondering how one can give a logical meaning to morphisms between prehilbertian spaces. If I was to consider such a morphism $f$ as a logical morphism between two $L$-structures, I should have ...
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47 views

Derive Fourier transform from what it should do?

I was wondering about the following: Imagine you want to figure out whether there is a transform that exchanges differentiation with multiplication and convoution with pairwise transformation for ...
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52 views

The trace class operators are the dual of the compact operators

I know that the map from the trace class operators $L_1(H)$ to the dual of the compact operators $K'(H)$ given by $A \mapsto tr( \cdot A)$ is an isometric isomorphism. Linearity is obvious by the ...