به انگلیسی

Measure vectors by testing them everywhere

Mathematics · Calculus & Analysis · ages 23-24

Name ______________________ Date ____________

For a nonzero bounded linear functional f on a normed space, what is its kernel?
- The entire space, including vectors with nonzero readings
- The closed hyperplane consisting of zero readings
- The single zero vector, regardless of the space's dimension
Hahn-Banach's norm-preserving extension theorem applies to complex normed spaces as well as real normed spaces.
Circle one: True False
On R^2 with the Euclidean norm, f(x,y)=5x+12y. What is the operator norm of f?
Answer: ______________
What does a linear functional on a vector space do?
- Maps each vector to a scalar linearly
- Maps each vector to a perpendicular vector
- Maps each nonzero vector to its positive length
Hahn-Banach's norm-preserving extension theorem applies only when the ambient normed space is complete.
Circle one: True False
On l1, the functional f_a(x)=sum a_n x_n has weights a=(1,-3,0.5,0,0,...). What is its operator norm?
Answer: ______________
In Euclidean R^2, which functional vanishes on the x-axis, takes value 1 at (0,4), and has norm 1/4?
- f(x,y)=y/4
- f(x,y)=y/2
- f(x,y)=y/8
- f(x,y)=2y
In finite-dimensional coordinates, which operation computes the pairing of a functional with a vector?
- Multiplying two column vectors entry by entry
- Multiplying the functional's row by the vector's column
- Adding the functional's row to the vector's transposed column
On R^n with the Euclidean norm, f(x)=a dot x. What is the operator norm of f?
- The number of coordinates in the vector a
- The number of zero coordinates in the vector a
- The Euclidean length of the vector a
Let v lie in the closure of a subspace M but not in M. Can a bounded linear functional vanish on M and be nonzero at v?
- Yes, because linearity prevents limits from affecting readings
- No, because continuity forces the reading at v to be zero
- Yes, because every subspace admits bounded separation outside it
- No, because every linear functional must vanish on every subspace

Answer key

For grown-ups. Fold this page away before handing over the rest.

Measure vectors by testing them everywhere W1-mt_pJuHFxkvjS-s1

The closed hyperplane consisting of zero readings · The kernel consists of vectors where f is zero. Since f is nonzero and linear, its kernel has codimension one. Continuity makes the kernel closed.
True · The extension theorem applies over either the real or the complex scalar field.
13 · The coefficient vector has Euclidean length sqrt(25+144)=13, which is the functional norm.
Maps each vector to a scalar linearly · A functional has scalar outputs and preserves addition and scalar multiplication. Length alone is not a linear functional.
False · Completeness is not a hypothesis of this extension theorem.
3 · For l1 inputs, the functional norm is the supremum of the absolute weights. That supremum is 3.
f(x,y)=y/4 · For f(x,y)=y/4, the x-axis readings are zero, f(0,4)=1, and the coefficient vector has length 1/4.
Multiplying the functional's row by the vector's column · A row times a column gives one scalar: the sum of matching coefficient-coordinate products.
The Euclidean length of the vector a · Cauchy-Schwarz bounds unit-vector readings by ||a||. The bound is attained in a's direction when a is nonzero.
No, because continuity forces the reading at v to be zero · Vectors of M can approach v. Continuity carries their zero readings to a zero reading at v.