My professor today remarked that the proof of the spectral theorem (even for the discrete spectrum case) uses not just algebra but also topology to prove the existence of eigenstates. However, I'm not being able to pinpoint which step of the proof of the spectral theorem (for infinite-dimensional operators) invokes topological arguments. Could someone please explain?
2026-03-28 22:29:53.1774736993
Where precisely is topology required to prove the existence of non-zero eigenstates in the proof of the spectral theorem?
78 Views Asked by user568976 https://math.techqa.club/user/user568976/detail At
1
There are 1 best solutions below
Related Questions in GENERAL-TOPOLOGY
- Is every non-locally compact metric space totally disconnected?
- Let X be a topological space and let A be a subset of X
- Continuity, preimage of an open set of $\mathbb R^2$
- Question on minimizing the infimum distance of a point from a non compact set
- Is hedgehog of countable spininess separable space?
- Nonclosed set in $ \mathbb{R}^2 $
- I cannot understand that $\mathfrak{O} := \{\{\}, \{1\}, \{1, 2\}, \{3\}, \{1, 3\}, \{1, 2, 3\}\}$ is a topology on the set $\{1, 2, 3\}$.
- If for every continuous function $\phi$, the function $\phi \circ f$ is continuous, then $f$ is continuous.
- Defining a homotopy on an annulus
- Triangle inequality for metric space where the metric is angles between vectors
Related Questions in FUNCTIONAL-ANALYSIS
- On sufficient condition for pre-compactness "in measure"(i.e. in Young measure space)
- Why is necessary ask $F$ to be infinite in order to obtain: $ f(v)=0$ for all $ f\in V^* \implies v=0 $
- Prove or disprove the following inequality
- Unbounded linear operator, projection from graph not open
- $\| (I-T)^{-1}|_{\ker(I-T)^\perp} \| \geq 1$ for all compact operator $T$ in an infinite dimensional Hilbert space
- Elementary question on continuity and locally square integrability of a function
- Bijection between $\Delta(A)$ and $\mathrm{Max}(A)$
- Exercise 1.105 of Megginson's "An Introduction to Banach Space Theory"
- Reference request for a lemma on the expected value of Hermitian polynomials of Gaussian random variables.
- If $A$ generates the $C_0$-semigroup $\{T_t;t\ge0\}$, then $Au=f \Rightarrow u=-\int_0^\infty T_t f dt$?
Related Questions in OPERATOR-THEORY
- $\| (I-T)^{-1}|_{\ker(I-T)^\perp} \| \geq 1$ for all compact operator $T$ in an infinite dimensional Hilbert space
- Confusion about relationship between operator $K$-theory and topological $K$-theory
- Definition of matrix valued smooth function
- hyponormal operators
- a positive matrix of operators
- If $S=(S_1,S_2)$ hyponormal, why $S_1$ and $S_2$ are hyponormal?
- Closed kernel of a operator.
- Why is $\lambda\mapsto(\lambda\textbf{1}-T)^{-1}$ analytic on $\rho(T)$?
- Show that a sequence of operators converges strongly to $I$ but not by norm.
- Is the dot product a symmetric or anti-symmetric operator?
Related Questions in SPECTRAL-THEORY
- Why is $\lambda\mapsto(\lambda\textbf{1}-T)^{-1}$ analytic on $\rho(T)$?
- Power spectrum of field over an arbitrarily-shaped country
- Calculating spectrum and resolvent set of a linear operator (General question).
- Operator with compact resolvent
- bounded below operator/ Kato-Rellich
- Show directly that if $E_1\geqslant E_2\geqslant\dots$, then $E_i\rightarrow \bigwedge E_i$ strongly.
- Is the spectral radius less than $1$?
- How to show range of a projection is an eigenspace.
- Spectral radius inequality for non-abelian Banach algebras
- Do unitarily equivalent operators have the same spectrum?
Related Questions in QUANTUM-MECHANICS
- Is there a book on the purely mathematical version of perturbation theory?
- Matrix differential equation and matrix exponential
- "Good" Linear Combinations of a Perturbed Wave Function
- Necessary condition for Hermician lin operators
- What is a symplectic form of the rotation group SO(n)
- Why is $\textbf{J}$ called angular momentum?(Quantum)
- How does the quantumstate evolve?
- Differential equation $au''(x)+b\frac{u(x)}{x}+Eu=0$
- How to model this system of $^{238}\,U$ atoms?
- Discrete spectra of generators of compact Lie group
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
At the very least you need to establish that the spectrum is nonempty. This usually requires the Fundamental Theorem of Algebra in the finite-dimensional case, or some form of Liouville's Theorem in the general case. Examples:
Theorem VII.3.6 in Conway's A Course in Functional Analysis.
Theorem 1.2.5 in Murphy's C$^*$-Algebras and Operator Theory
Theorem 3.2.3 in Kadison-Ringrose's Fundamentals of Theory of Operator Algebras
Theorem 4.1.13 in Pedersen's Analysis Now
Lemma VII.3.4.4 in Dumford-Schwartz Linear Operators