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Questions tagged [functional-analysis]

Functional analysis, the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, including spectral theory, as well as measure, integration, probability on infinite dimensions, and also manifolds with local structure modeled by these vector spaces.

For basic questions about functions use more suitable tags like , or .

52582 questions
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Compact Hermitian Operator, closed range implies finite rank

Let $K$ be a compact Hermitian operator on a Hilbert space $H$. How to show that if $K$ has infinite rank then the range of $K$ is not closed in $H$? My thought is that suppose range $K$ is closed in $H$ and then we need to prove that $K$ has…
Carlotto
  • 101
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Sturm-Liouville Problem

How could one prove that there are at most countably many eigenvalues of the Sturm-Liouville problem $-Lu = ju$, $j$ = eigenvalue, and $u$ is in $C^2[a,b]$? I have been attempting at this problem for a while and I am not sure how to proceed. Thank…
Carlotto
  • 101
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2 answers

Norm of functional on $C[-1,1]$ defined as $P(f) = f(1)+f(-1)-2f(0)$

I am studying for my final exam and kinda struggling with the following: Calculate the norm of functional on $C[-1,1]$ defined as $P(f) = f(1)+f(-1)-2f(0)$ This seems like one-liner but I am clueless anyway. I appreciate your time.
blahblah
  • 2,160
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2 answers

Differentiable Operator Continuous?

Consider the space $C^{\infty}[a,b]$ with norm $||f|| = max_{[0,1]} |f(x)|$, with $f\in C^{\infty}[a,b]$. Is the differentiation operator $\frac{d}{dx}$ continuous on $C^{\infty}[a,b]$? I'm very confused because it seems almost trivial-- there are…
user79015
  • 31
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Is the space of smooth functions on a noncompact manifold nuclear?

I have two questions, really: Q1) Suppose $M$ is a noncompact, finite-dimensional smooth manifold. Is it true that $C^{\infty}(M)$, by which I will always mean $C^{\infty}(-; \mathbb{R})$, is a nuclear Frechet space? Reason for asking: if you…
GuestAccount99
  • 31
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Incomplete space

I must show that $C_0(\mathbb{R})$ space of all continuous real functions $f:\mathbb{R} \longrightarrow \mathbb{R}$ with compact support is not a complete space endowed by norm $\|f\|=\sup\limits_{t \in \mathbb{R}}|f(t)|$. Thank for any help.
Charles Seife
  • 563
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Exercise 4 Conway, a course in functional analysis.

I am trying to test Conway problem 4 page 49. I'm having trouble proving an implication. Corollary 5.4. If $T$ is a compact self-adjoint operator, then there is a sequence $\left\{\mu_n\right\}$ of real numbers and an orthonormal basis…
eraldcoil
  • 3,508
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1 answer

Closure of unitaries in the SOT

Let $H$ be a Hilbert space. I am trying to find the closure of the collection of unitary operators $U(H)$ in $L(H)$ in SOT. I know the strong limit of unitary operators isnt unitary (Unitary shifts on $\ell^2$). I cannot find any information on this…
user515599
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1 answer

$T_f:L^2[0,1] \to L^2[0,1]$ defined as $T_f=fg$ is bounded and compact only if $f=0$.

I am trying to solve the following problem: Let $f:[0,1] \to \mathbb{C}$ be a continuous function and $T_f:L^2[0,1] \to L^2[0,1]$ be the operator given by $T_f(g)=fg, \; g \in L^2[0,1].$ Prove that $T_f$ is a bounded linear operator on $L^2[0,1]$…
mat95
  • 339
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Extending linear continuous operator on subspace of $L^\infty$?

Consider $S$ to be the subspace of $L^\infty(\mathbb R)$ consisting of all functions supported on finite measure sets and $T:S\to B$ is a linear continuous operator between $S$ and some Banach space $B$, is there always a linear continuos mapping…
stephenkk
  • 977
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The Subspace $M=\{f\in C[0,1]:f(0)=0\}$

Let $C[0,1]$ with the supremum norm. It's easy to see that $M=\{f\in C[0,1]:f(0)=0\}$ is a closed subspace and so $C[0,1]/M$ is a Banach space. But I'm having trouble in finding a Banach space isometric to $C[0,1]/M$.
user34870
  • 1,173
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Bounded sequence in a Hilbert space has weakly convergent subsequence

Suppose $H$ is a Hilbert space with an orthonormal basis $(e_n)$. Let $(x_n)$ be a bounded sequence in $H$. Prove that $(x_n)$ contains a subsequence $(x_{n_k})$ and $H$ contains an element $x$ so that $\langle x_{n_k},e_n\rangle\to\langle…
autumnriddle
  • 33
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$0\leq a \leq b $ implies $ b^{-1} \leq a^{-1}$ in C*-algebra?

I'm studying C*-algebra in Conway's Functional Analysis. Problem is this. $0\leq a \leq b $, then $ b^{-1} \leq a^{-1}$ in C*-algebra? I tried this problem via this way. 1st trial) Since $ a^{-1} - b^{-1} = a^{-1}(b-a)b^{-1}$ so product of three…
probafds123
  • 402
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1 answer

Why is the derivative operator never bounded?

Consider $ X= C^\infty ([0,1],\mathbb{R} )$ and the operator $$ D: (X,\|\cdot\| ) \to (X,\|\cdot\|).$$ given by $$D(x)=x'.$$ (derivation operator). Why is unbounded, independently of the choice of norm? I can prove this for cases where I know the…
Ashkanani
  • 99
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1 answer

Calculating average over a function set

Non-math version of the problem: I am running with a GPS device, recording my path. I know the curve the GPS has recorded. However, the GPS device actually has an accuracy, which I can assume to be fixed (for starters). Assuming that the curve the…
K.Steff
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