Explore tens of thousands of sets crafted by our community.
Brouwer Fixed Point Theorem
5
Flashcards
0/5
The theorem guarantees at least one fixed point in any continuous function from a compact convex set to itself in a Euclidean space.
True, this is the general statement of the Brouwer Fixed Point Theorem.
The Brouwer Fixed Point Theorem implies that a continuous function mapping a sphere to itself must have a fixed point.
False, the theorem applies to closed disks, not spheres. A sphere is a boundary of a closed disk, without the interior points.
The Brouwer Fixed Point Theorem applies to any function from the real numbers to the real numbers.
False, the theorem specifically applies to continuous functions from a closed disk to itself within Euclidean spaces.
Brouwer Fixed Point Theorem can be applied in spaces other than Euclidean, such as in a Banach space.
False, the theorem is specific to finite-dimensional Euclidean spaces and does not generalize to infinite-dimensional Banach spaces.
The Brouwer Fixed Point Theorem can be deduced using algebraic topology, for instance, with the use of homology or cohomology theories.
True, algebraic topology provides tools like homology that can be used to prove the Brouwer Fixed Point Theorem.
© Hypatia.Tech. 2024 All rights reserved.
