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

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Prob. 10, Sec. 3.10 in Kreyszig's functional analysis book: Every isometric linear operator on a finite-dimensional inner product space is unitary? [duplicate]

Let $X$ be an inner product space such that $\dim X < \infty$, and let $T \colon X \to X$ be an isometric linear operator. Since $\dim X < \infty$, $X$ is complete and thus a Hilbert space; ...
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43 views

Prob. 9, Sec. 3.10 in Kreyszig's functional analysis book: The image of ann isometric non-unitary operator on a Hilbert space

Let $H$ be a Hilbert space, let $T \colon H \to H$ be a linear operator such that $T$ is isometric but not unitary. Then how to show that the image $T[H]$ is a proper closed subspace of $H$? My ...
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41 views

Show that if a vector subspace of a Hilbert space is closed, then it is a Hilbert subspace.

Show that if a vector subspace of a Hilbert space is closed, then it is a Hilbert subspace. Here is what I have so far. Any comments or hints are greatly appreciated. Let $H$ be a Hilbert space and ...
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39 views

Find an inner product that makes a given set of linearly independent vectors orthogonal

I need to find an inner product such that given a set $S$ of linearly independent vectors in a Hilbert space $H$, $S$ will be orthogonal with these product. I thought Gram -Schmidt Process would help ...
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31 views

$Az + B\overline{z}$ as a linear operator

Given two matrices $A,B \in \mathbb{C}^{n\times n}$ with fixed $n\in\mathbb{N}^+$, let us consider the operator $$ L:\mathbb{C}^n \to \mathbb{C}^n,\\ L(z) = Az + B\overline{z}. $$ This operator is not ...
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38 views

Exotic applications of Hilbert spaces?

So my final exam for an introductory course on Hilbert spaces is just a weeks away. I enjoyed the course, we covered the theory in enough detail to illustrate its richness and elegance. I'm aware of ...
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75 views

Prove $\sin(kx) \rightharpoonup 0$ as $k \to \infty$ in $L^2(0,1)$ [duplicate]

I want to show that $u_k(x)= \sin(kx) \rightharpoonup 0$ as $k \to \infty$ in $L^2(0,1)$. We know trivially that $0 \in L^2(0,1)$. I need to show that $\langle u^*,\sin(kx) \rangle \to \langle ...
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64 views

Show that there exists no positive continuous function $f$ defined on $[a,b]$ that satisfies the following conditions:

Show that there exists no positive continuous function $f$ defined on $[a,b]$ that satisfies the following conditions: $\int_a^bf(t)dt=1$, $\int_a^btf(t)dt=\alpha$, $\int_a^bt^2f(t)dt=\alpha^2$ ...
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37 views

Subsec. 4.1-8 in Kreyszig's functional analysis book: Does every inner product have a total orthonormal set?

In every Hilbert space $H \neq \{0 \}$, there exists a total orthonormal set. I think I've understood the proof given by Erwin Kreyszig in Introductory Functional Analysis With Applications. ...
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53 views

An inner product on the dual space of a non-complete inner product space?

As is well known, for any Hilbert space $V$, there is a natural inner product on the continuous dual. (the space of all continuous linear functionals). Is there a way to endow an inner product on ...
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62 views

Prob. 10, Sec. 3.9 in Kreyszig's functional analysis book: The null space and adjoint of the right-shift operator

Let $(e_n)$ be a total orthonormal sequence in a separable Hilbert space $H$, let $T \colon H \to H$ be defined as follows: Since span of $(e_n)$ is dense in $H$, for every $x \in H$, we have $$x = ...
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25 views

Prove that matrix $[S]$ associated to operator is such that $A |\zeta|^2\leq s_{ij}(x) \zeta_i \zeta_j\leq B |\zeta|^2$.

Let us consider $N\times N$ matrix $[S]$ associated to operator $S:V\rightarrow V$ where $V$ is a Hilbert space; $S$ is linear, bounded, invertible, positive and self-adjoint. Prove that $[S]$ is ...
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61 views

Prob. 4, Sec. 3.9 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Image of a set under the adjoint operator

Here's Prob. 4, Sec. 3.9 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $H_1$ and $H_2$ be Hilbert spaces, and let $T \colon H_1 \to H_2$ be a bounded linear ...
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23 views

Showing that an operator is bijective

Assume that $ A $ generates a contraction semigroup on a Hilbert space $ X $, and B is a bounded linear operator on $ X $. I want to show that $ A + B - 2|| B ||I $ with the domain equal to the domain ...
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28 views

Prob. 2, Sec. 3.9 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Inversion and adjointness

Let $H$ be a Hilbert space, and let $T \colon H \to H$ be a bijective bounded linear operator whose inverse is bounded. Then how to show that $(T^*)^{-1}$ exists and $$(T^*)^{-1} = (T^{-1})^*?$$ My ...
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58 views

Classical solution involving semigroups

Let $\{T_{D}(t)\}_{t\ge 0}$ a $C_{0}$-semigroup with generator $A+D$, with $D\in\mathcal{L}(X)$ and let $x_{0}\in \mathcal{D}(A)$. I want to show that ...
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64 views

Prob. 7, Sec. 3.8 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: The dual space of a Hilbert space is a Hilbert space.

Here's Prob. 7, Sec. 3.8 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Show that the dual space $H^\prime$ of a Hilbert space $H$ is a Hilbert space with inner product ...
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36 views

$\overline{\mathrm{Im} (T^*T)} = \overline{\mathrm{Im} T^*}$

I need to prove that in a Hilbert space, $\overline{\mathrm{Im}(T^*T)} = \overline{\mathrm{Im}T^*}$. I have already shown that $\ker (T^*) = (\mathrm{Im} T)^\perp$ and have so far concluded that ...
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48 views

Theorem 3.8-1 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Do we really need the completeness of the space?

Here's Theorem 3.8-1 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Every bounded linear functional $f$ on a Hilbert space $H$ can be represented in terms of the inner ...
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28 views

Biorthogonal complement of subspace of subspace.

I'm taking a course on Banach and Hilbert spaces. The teacher who guides the exercise sessions is often a bit fast, so only when revising my notes at home I realize I do not fully understand them. We ...
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34 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 ...
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78 views

Example of projection sequence on Hilbert space with strong limit P

Let $P_n$ be strongly convergent with limit $P$, where $P_n$'s are projections on a Hilbert space $H$.Suppose that $P_n(H)$ is infinite dimensional. Show by example that P(H)$ may be finite ...
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145 views

Spectral theorem for a pair of commuting operators

Let $H$ be Hilbert space and $A$, $B$ - self-adjoint (bounded or unbounded) operators on $H$. According to spectral theorem for every bounded Borel function $f: \mathbb{R}\to \mathbb{R}$ we have ...
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21 views

Cauchy sequence of partial sums of orthogonal vectors in a general Hilbert Space.

Let $(x_n)$ be a sequence of orthogonal vectors in a Hilbert space $(V, \langle,\rangle)$. For $n = 1, 2, 3, ... $ put, $$s_n = \sum_{j=1}^{n} x_j.$$ (a) Calculate $\|s_n\|$ in terms of ...
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45 views

Show $\lim_{n\to\infty}(Lu_n) = L(\lim_{n\to\infty} u_n)$

Suppose $\{u_n\}$ is a convergent sequence in Hilbert space $H$ and $L$ is a bounded (continuous) linear operator on $H$. Use the definition of convergence to show that $\lim_{n\to\infty}(Lu_n) = ...
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117 views

Homeomorphism on the Hilbert space

We can consider two different topologies on the Hilbert space ; $l^{2}(\mathbb{N})$. One is the topology deduced from the norm \begin{equation*} \|f\|=\sqrt{\sum_{n=1}^{\infty} f(n)^{2}}, ...
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27 views

Orthogonal set of a set in Hilbert space

This is an exercise in the Folland Real Analysis p.177. I first thought it is an easy one, but it turns out to be a lot trickier..... I have no idea how to deal with the so-called "double ...
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54 views

Countable set in a Banach space which spans densely?

Let $\mathcal{C}(\mathbf{T})$ be the algebra of continuous complex functions on the unit circle $\mathbf{T}$. Consider the following two statements: The $*$-subalgebra generated by $1$ and $z$ spans ...
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20 views

unique inner product on a tensor product of Hilbert $C^*$ modules and Hilbert spaces.

For a $C^*-$ algebra $A$ and a Hilbert space $H$ and a Hilbert $A-$module E; how can we show that there is a unique $A-$ valued inner product on $H \otimes E$ as $< h_1 \otimes x_1 , h_2 \otimes ...
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91 views

State of a $ C^{*} $-algebra.

Let $ (\pi,\mathcal{H}) $ be a non-degenerate $ * $-representation of a $ C^{*} $-algebra $ A $, and let $ h \in \mathcal{H} $ with $ \| h \| = 1 $. Define $ f_{h}: A \to \Bbb{C} $ by $ {f_{h}}(a) ...
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22 views

$A$ be a non-empty closed convex subset of a Hilbert space $H$ , is the distance from $A$ always attained at a unique point in $A$ ?

Let $A$ be a non-empty closed convex subset of a Hilbert space $H$ , then is it true that for every $b \in H$ , $\exists$ unique $x_b \in A$ such that $||x_b-b||=d(b,A)=\inf \{||b-x||:x \in A\}$ ?
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34 views

Norm is sup of inner products (proof).

Let $V$ be a vector space with an inner product $\langle.,. \rangle$ and associated norm $|| . ||$ Then: Could I have a proof of this fact?
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28 views

On the proof of the continuity of the inner product.

I am having problems with the following proof and I need to fill in some details: I understand that continuity is being proven by the sequence definition but I do not get why (a) follows ...
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1answer
157 views

Lemma 3.5-3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Is the set of non-zere Fourier co-efficients uncountable too?

Let $X$ be an inner product space, let $x \in X$ be non-zero, and let $M$ be an uncountable orthonormal subset of $X$. Then what can we say about the cardinality of the following set? $$ \{ \ v \in M ...
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22 views

What can one assume about $T^*$ when showing that $T$ is normal?

Consider a continuous and linear operator $T$ such that $$ T : l^2 \to l^2 $$ where $(a_n) \mapsto (\alpha_n a_n)$ Moreover $(\alpha_n)$ is a sequence of complex numbers that converges to zero. Now, ...
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65 views

Isolated Eigenvalue

What does it mean that an eigenvalue is "isolated"? My intuitive understanding says it is when one can find an open ball around it such that there is no other eigenvalue in that open ball. However, I ...
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27 views

Norm of operator matrix

I'm having trouble with the following: suppose H is a Hilbert space and $f_{i, j}, g_{i, j} : H \rightarrow H$, $1 \leq i, j \leq n$ are bounded operators. Then we have operators $(f_{i, j}) , (g_{i, ...
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49 views

Hilbert space L2 - inner product

I have a problem with one exercise. I have to prove that $L^2$ space is Hilbertian. So I think that the best way is to check out inner product by definition of norm, so: \begin{equation*} ...
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1answer
99 views

Showing that an operator generates a contraction semigroup

Let $A$ be the infinitesimal generator of a contraction semigroup $(T(t))_{t\ge 0}$ on the Hilbert space $X$, and $D\in\mathcal{L}(X)$. I want to show that the operator $A+D-2\|D\|I$ with domain ...
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36 views

Prove Operator is a Projector

Let $\mathscr{H}$ be a complex Hilbert space. A projector is a linear map $P:\mathscr{H}\to\mathscr{H}$ such that $P\circ P = P$. I'm trying to prove the following claim, from the information given ...
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30 views

Is this a metric on $\mathbf P\mathcal H$?

Let $\mathcal H$ be a real or complex Hilbert space with inner product $\langle\cdot,\cdot\rangle$. On the projective space $\mathbf P\mathcal H = \left(\mathcal H\setminus\{0\}\right)\big/{\sim}$ ...
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63 views

Powers of compact operators

Consider a Hilbert space $H$ and a compact self-adjoint operator $T : H \to H$. I want to prove that all positive powers (especially fractional powers) of $T$ are compact. From the spectral theorem, I ...
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34 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 ...
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27 views

On the subset of a closed vector subspace

Theorem: Let $H$ be a Hilbert space, and let $U$ and $V$ be closed subspaces of $H$ such that $U\subset V$. Then there exists a nonzero vector $v\in V\backslash U$ such that $v\bot U$. The fact that ...
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36 views

Need help understanding compact embedding of hilbert spaces

I am trying to understand the following statement, and I would like some clarification Consider a Hilbert space $H$ which is compactly embedded in a Hilbert space $L$, with $H^*$ being the dual ...
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20 views

Norm of a linear continuous form

Let $E=\{f\colon[0,2]\to\mathbb{R} \mid f \text{ continuous} \}$ be a prehilbert space equipped with inner product: $$\langle f,g\rangle=\int_0^2 f(t)g(t)\, dt$$ And let : $$U\colon E ...
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24 views

Proving an integral identity

I'm dealing with the Hermitian operator, and I've been asked to prove that all $f(x) = x^n e^{\alpha x}$ belong to $L^2(-\infty,\infty;e^{-x^2/2})$ by showing that: $$\int_{-\infty}^{\infty}x^m ...
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52 views

Intersection of Hilbert spaces

Consider two Hilbert spaces $H_1$ and $H_2$ with inner products $\langle \cdot,\cdot\rangle_1$ and $\langle \cdot,\cdot\rangle_2$ generating norms $\Vert \cdot \Vert_1$ and $\Vert \cdot \Vert_2$ ...
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30 views

Theorem 3.3-1, Lemma 3.3-2, and Theorem 3.3-4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: How to write these as one?

I'm trying to prepare some ancilliary material on the following three results in sec. 3.3 in the book Introductory Functional Analysis With Applications by Erwine Kreyszig: (First, I'm giving ...
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51 views

Prob. 2, Sec. 3.3 in Erwin Kreyszig's “Introductory Functional Analysis With Applications”: How to minimise the norm?

Let $z$ be a given complex number. Let $M \subset \mathbb{C}^n$ be given by $$M \colon= \left\{ (\xi_1, \ldots, \xi_n ) \in \mathbb{C}^n \mid \sum_{i=1}^n \xi_i = z \right\}.$$ Then $M$ is convex ...