- 11.4.1: Show that the functions d1 and doodefined in 11.4.2(c) are metrics ...
- 11.4.2: Show that the functions dooand d1 defined in 11.4.2(d, e) are metri...
- 11.4.3: Verify that the discrete metric on a set S as defined in 11.4.2(f) ...
- 11.4.4: If Pn:= (xn' Yn) e JR2anddoois themetricin 11.4.2(c),showthat(Pn)co...
- 11.4.5: Verify the conclusion of Exercise 4 if doois replaced by d1.
- 11.4.6: Let S be a nonempty set and let d be the discrete metric defined in...
- 11.4.7: Show that if d is the discrete metric on a set S, then every subset...
- 11.4.8: Let P := (x, y) and := (0,0) in JR2.Draw the following sets in the ...
- 11.4.9: Prove that in any metric space, an s-neighborhood of a point is an ...
- 11.4.10: Prove Theorem 11.4.11.
- 11.4.11: Prove Theorem 11.4.12.
- 11.4.12: Prove Theorem 11.4.12.If (S, d) is a metric space, a subset A S; S ...
Solutions for Chapter 11.4: Metric Spaces
Full solutions for Introduction to Real Analysis | 3rd Edition
Additive inverse of a complex number
The opposite of a + bi, or -a - bi
A relationship between two variables in which the values of the response variable are directly affected by the values of the explanatory variable
See Linear programming problem.
Exponential decay function
Decay modeled by ƒ(x) = a ? bx, a > 0 with 0 < b < 1.
Focal length of a parabola
The directed distance from the vertex to the focus.
Imaginary part of a complex number
See Complex number.
Integrable over [a, b] Lba
ƒ1x2 dx exists.
Zeros of a function that are irrational numbers.
Mode of a data set
The category or number that occurs most frequently in the set.
See Principal nth root
An annuity in which deposits are made at the same time interest is posted.
Plane in Cartesian space
The graph of Ax + By + Cz + D = 0, where A, B, and C are not all zero.
A relationship between two variables in which higher values of one variable are generally associated with higher values of the other variable, p. 717.
Projection of u onto v
The vector projv u = au # vƒvƒb2v
A procedure for fitting a quartic function to a set of data.
Reflection across the y-axis
x, y and (-x,y) are reflections of each other across the y-axis.
If a polynomial f(x) is divided by x - c , the remainder is ƒ(c)
a - b = a + (-b)
A line graph in which time is measured on the horizontal axis.
A function that maps real numbers to real numbers.